1. bun15383kh@gmail.com sent to www.laisac.page.tlTUYN CHN CC BI TON PHNG TRNH, H PHNG TRNH, BT PHNG TRNH TRONG THI HC SINH GII CC TNH, THNH PH NM HC 2010 - 2011 (L Phc L - tng hp v gii thiu)Bi 1. 1/ Gii phng trnhx 2 x 1 x 3 4 x 1 1 .2/ Gii phng trnh vi n s thc 1 x 6 x 5 2 x ( thi HSG tnh Vnh Long) 5432Bi 2. Gii phng trnh x x x 11x 25 x 14 0 ( thi HSG tnh ng Nai) 2x 2 y 4 Bi 3. Gii h phng trnh 2x 5 2y 5 6 ( HSG B Ra Vng Tu) 1 x x y 3 3 y Bi 4. Gii h phng trnh sau 2 x y 1 8 y ( thi HSG Hi Phng, bng A)4 x 2 y 4 4 xy 3 1 Bi 5. Gii h phng trnh 2 2 4 x 2 y 4 xy 2 ( thi HSG tnh Lm ng)x4 5 y 6 Bi 6. Gii h phng trnh trn tp s thc 2 2 x y 5x 6 ( thi chn i tuyn ng Nai) 1

2. 3 2y x2 y 2 1 x 1 Bi 7. Gii h phng trnh x2 y2 2x 4 y ( thi HSG H Tnh) Bi 8. Gii phng trnh3x 6 x2 7 x 1 ( thi chn i tuyn Lm ng) x x y 1 1 Bi 9. Gii h phng trnh 2 2 y x 2y x y x 0 ( thi HSG tnh Qung Bnh) Bi 10. 1/ Gii bt phng trnh ( x 2 4 x) 2 x 2 3 x 2 0 . xy y 2 x 7 y 2/ Gii h phng trnh sau x 2 y x 12 ( thi HSG in Bin) 8 10 6 x y z 1 . Bi 11. Gii h bt phng trnh 2007 2009 2011 x y z 1 ( thi chn i tuyn Bnh nh) Bi 12. 1/ Gii phng trnhx 1 1 x 2 x 1 3 xx2 x 2 y 2/ Gii h phng trnh 2 y y 2x ( thi HSG tnh Bn Tre) 2 3. Bi 13. 1/ Gii phng trnh x 2 4 x 3 x 5 . 2/ Gii phng trnh x 3 x 2 3x 1 2 x 2 trn [2, 2] ( thi HSG tnh Long An) 1 y 2 x 2 y Bi 14. Gii h phng trnh sau x x 2 2 y ( x 1 1) 3 x 3( chn i tuyn trng Chuyn L Qu n, Bnh nh).2 x 2 y 3 xy 4 x 2 9 y Bi 15. Gii h phng trnh sau 2 7 y 6 2 x 9 x ( thi chn i tuyn Nha Trang, Khnh Ha) Bi 16. 1/ Gii phng trnh x 2 7 x 2 x 1 x 2 8 x 7 1 2 2 x y 3 2 x y 2/ Gii h phng trnh 3 x 6 1 y 4 ( thi HSG tnh Vnh Phc) Bi 17. Gii phng trnh sau x 4 2 x 3 2 x 2 2 x 1 ( x 3 x)1 x2 x ( thi HSG tnh H Tnh)Bi 18. Gii phng trnh 2 sin 2 x 3 2 sin x 2 cos x 5 0 . ( thi chn i tuyn trng THPT chuyn L Khit, Qung Ngi) Bi 19. 1/ Gii phng trnh x 4 x 2 2 x 4 x 2 . 3 4. 2 y ( x 2 y 2 ) 3x 2/ Gii h phng trnh 2 2 x( x y ) 10 y ( thi chn i tuyn THPT Chuyn Lam Sn, Thanh Ha) Bi 20. Gii phng trnh3x 6 x2 7 x 1 . ( thi HSG tnh Lm ng) 5( x y ) 6( x z ) x y 6 xy x z 5 xz 4 6( z y ) 4( x y ) Bi 21. Gii h phng trnh 5 z y 4 zy x y 6 xy 4( x z ) 5( y z ) 6 x z 5 xz y z 4 yz ( chn i tuyn trng PTNK, TPHCM) Bi 22. 1/ Gii phng trnhx 2 y 1 3 z 2 1 ( x y z 11) 2x 2 121 2 27 x 2x 2/ Gii h phng trnh 9 2 2 x y xy 3x 4 y 4 0( thi HSG tnh Qung Nam) Bi 23. 1/ Tm tt c cc gi tr ca a, b phng trnhx 2 2ax b m c hai nghim phn bit vi bx 2 2ax 1mi tham s m. y xy 2 6 x 2 2/ Gii h phng trnh 3 3 3 1 x y 19 x ( thi HSG vng tnh Bnh Phc) Bi 24. 4 5. x 2 y 2 z 2 2010 2 1/ Gii h phng trnh 3 3 3 3 x y z 2010 2/ Gii phng trnh 32 x3 x2 3x32x x3 3x 2 0 ( thi chn i tuyn Ninh Bnh)Bi 25. 2 x y x 1 2( x 1) 2(2 x y ) 2 1/ Gii bt phng trnh sau 2 y 4 x x 1 17 0 2/ Vi n l s nguyn dng, gii phng trnh1 1 1 1 ... 0. sin 2 x sin 4 x sin 8 x sin 2 n x( thi HSG tnh Khnh Ha) Bi 26. 1/ Gii phng trnh sau2/ Gii phng trnh log 33 sin 2 x cos 2 x 5sin x (2 3) cos x 3 3 1. 2 cos x 32x 1 3x2 8x 5 2 ( x 1) ( thi HSG tnh Thi Bnh)Bi 27. x 2 1 y 2 xy y 1/ Gii h phng trnh y x y 2 1 x2 2/ Gii phng trnh lng gic2 2 2 2 sin 2 x tan x cot 2 x ( thi HSG tnh Ph Th)Bi 28. Gii phng trnh 24 x 2 60 x 36 1 1 0 5x 7 x 1 ( thi HSG tnh Qung Ninh) 5 6. Bi 29. Gii phng trnh3 x 3 2 x 2 2 3 x 3 x 2 2 x 1 2 x 2 2 x 2 ( thi chn i tuyn trng THPT Chuyn HSP H Ni)(2 x 2 3 x 4)(2 y 2 3 y 4) 18 Bi 30. Gii h phng trnh 2 2 x y xy 7 x 6 y 14 0 ( thi chn i tuyn trng THPT Chuyn HSP H Ni) 2(2 x 1)3 2 x 1 (2 y 3) y 2 Bi 31. Gii h phng trnh 4x 2 2y 4 6 ( thi chn i tuyn trng THPT chuyn Lng Th Vinh, ng Nai) x 4 x3 y 9 y y 3 x x2 y 2 9 x Bi 32. Gii h phng trnh 3 3 x( y x ) 7 ( thi chn HSG tnh Hng Yn) 2 y 3 2 x 1 x 3 1 x y Bi 33. Gii h phng trnh 2 y 2 x 1 2 xy 1 x ( thi chn i tuyn chuyn Nguyn Du, k Lk) x 3 y 3 35 Bi 34. Gii h phng trnh 2 2 2 x 3 y 4 x 9 y ( thi HSG tnh Yn Bi) Bi 35. Gii phng trnh 2 3 2 x 1 27 x 3 27 x 2 13x 2 ( thi HSG Hi Phng, bng A1)1 1 2 2 x 2 y 2( x y ) Bi 36. Gii h phng trnh 1 1 y2 x2 x 2y ( thi chn i tuyn Qung Ninh) 6 7. x 3 3 x 12 y 50 Bi 37. Gii h phng trnh y 3 12 y 3 z 2 z 3 27 x 27 z ( thi chn i tuyn trng THPT Phan Chu Trinh, Nng)Bi 38. Gii phng trnh3x9 9 x 2 1 2x 1 3 ( thi chn i tuyn Ph Yn)Bi 39. 1/ Gii phng trnh sau x 1 x 1 2 x x 2 2 y 3 y x 3 3x 2 4 x 2 2/ Gii h phng trnh sau 2 1 x y 2 y 1 ( thi HSG tnh Ngh An) Bi 40. x 3 8 y 3 4 xy 2 1 1/ Gii h phng trnh 4 4 2 x 8 y 2 x y 0 2/ Chng minh phng trnh sau c ng mt nghim ( x 1) 2011 2( x 1) 3 x 3 3 x 2 3 x 2 . ( d b thi HSG tnh Ngh An) x 3 y x 3 12 Bi 41. Gii h phng trnh sau y 4 z y 3 6 3 9 z 2 x z 32( thi chn i tuyn KHTN, vng 1) y 2 x2 x 2 1 2 e Bi 42. Gii h phng trnh y 1 3 log ( x 2 y 6) 2 log ( x y 2) 1 2 2 7 8. ( thi chn i tuyn trng THPT Cao Lnh, ng Thp) Bi 43. Gii phng trnh saux2 x 2 2x2 x 2 x2 11 x x 2 1 x x 4 ( thi HSG tnh Bnh Phc) Bi 44. 1/ Gii phng trnh33x 4 x3 3x 2 x 22/ Tm s nghim ca phng trnh (4022 x 2011 4018 x 2009 2 x) 4 2(4022 x 2011 4018 x 2009 2 x) 2 cos 2 2 x 0 ( thi chn i tuyn Chuyn Nguyn Du)(2 x)(1 2 x )(2 y )(1 2 y ) 4 10 z 1 Bi 45. Gii h phng trnh sau 2 2 2 2 2 x y z 2 xz 2 yz x y 1 0 ( thi chn i tuyn H Tnh) Bi 46. 1/ Gii phng trnh sau 2010 x ( x 2 1 x) 1 . y 4 4 x 2 xy 2 x 4 5 2/ Gii h phng trnh x 3 3 y 2 x y 2 ( thi chn i tuyn trng THPT So Nam, tnh Qung Nam) x11 xy10 y 22 y12 Bi 47. Gii h phng trnh 4 4 2 2 7 y 13x 8 2 y 3 x(3x 3 y 1) ( thi chn i tuyn TP.HCM) 2009 x 2010 y ( x y ) 2 Bi 48. Gii h phng trnh 2010 y 2011z ( y z ) 2 2 2011z 2009 x ( z x )( thi chn i tuyn chuyn Quang Trung, Bnh Phc) 8 9. 1 2 2 x y 5 Bi 49. Gii h phng trnh sau 4 x 2 3 x 57 y (3 x 1) 25 ( thi chn i tuyn Ngh An) Bi 50. Cho cc tham s dng a, b, c . Tm nghim dng ca h phng trnh sau : x y z a b c 2 2 2 4 xyz a x b y c z abc( kim tra i tuyn Ninh Bnh)3x y x x2 y 2 3 Bi 51. Gii h phng trnh sau trn tp hp s thc y x 3y 0 x2 y 2 ( thi chn i tuyn Chuyn Vnh Phc, tnh Vnh Phc) 4 4 x 2x y y Bi 52. Gii h phng trnh 2 2 3 ( x y ) 3 ( kim tra i d tuyn trng THPT Chuyn HSP H Ni) Bi 53. Gii phng trnh 2 x 2 .sin x x.cos x 3 2 x 1 x 3 x 5 x 1 ( thi chn i tuyn H Ni) ( x 2)2 ( y 3) 2 ( y 3)( x z 2) Bi 54. Gii h phng trnh x 2 5 x 9 z 7 y 15 3 yz 2 2 8 x 18 y 18 xy 18 yz 84 x 72 y 24 z 176 ( thi chn i tuyn HSP H Ni, ngy 2)Bi 55. 2 z ( x y ) 1 x 2 y 2 Tm x, y, z tha mn h y 2 z 2 1 2 xy 2 zx 2 yz 2 2 y (3x 1) 2 x ( x 1)( thi chn i tuyn trng H KHTN H Ni, vng 3)9 10. LI GII CHI TIT V NHN XT Bi 1. 1/ Gii phng trnhx 2 x 1 x 3 4 x 1 1 .2/ Gii phng trnh vi n s thc 1 x 6 x 5 2 x ( thi HSG tnh Vnh Long) Li gii. 1/iu kin x 1 . Phng trnh cho tng ng vi ( x 1 1)2 ( x 1 2) 2 1 -Nux 1 1 x 1 2 1 (*)x 1 1 th (*) ( x 1 1) ( x 1 2) 1 3 2 x 1 1 x 1 1 , loi.-Nu 1 x 1 2 2 x 5 th (*) ( x 1 1) ( x 1 2) 1 1 1 , lun ng. -Nux 1 2 th (*) ( x 1 1) ( x 1 2) 1 2 x 1 3 1 x 1 2 , loi.Vy phng trnh cho c nghim l mi x thuc 2;5 . 2/ iu kin x 5 . Phng trnh cho tng ng vi 21 x 5 2 x 6 x (1 x ) (5 2 x ) 2 (1 x)(5 2 x ) 6 x (1 x )(5 2 x ) x 5 (1 x)(5 2 x) x 2 10 x 25 x 2 7 x 30 0 x 3 x 10 Th li, ta thy ch c x 3 l tha mn.Vy phng trnh cho c nghim duy nht l x 3 . Nhn xt. Cc dng ton phng trnh v t ny kh c bn v quen thuc, chng hon ton c th gii bng cch bnh phng kh cn m khng cn lo ngi v tnh gii c ca phng trnh hay khng. n gin trong vic xt iu kin, ta c th gii xong ri th li cng c. 10 11. Bi 2. Gii phng trnh x 5 x 4 x 3 11x 2 25 x 14 0 ( thi HSG tnh ng Nai) Li gii. Phng trnh cho tng ng vi ( x 5 2 x 4 ) ( x 4 2 x3 ) ( x3 2 x 2 ) ( 9 x 2 18 x) (7 x 14) 0 ( x 2)( x 4 x 3 x 2 9 x 7) 0 x 2 4 3 2 x x x 9x 7 0 Phng trnh th hai trn c th vit li l ( x 4 x3 x 2 9 x 6) 1 0 ( x 4 x 3 2 x 3 2 x 2 3x 2 3 x 6 x 6) 1 0 ( x 1)2 ( x 2 3 x 6) 1 0Do ( x 1)2 ( x 2 3 x 6) 1 0, x nn phng trnh ny v nghim. Vy phng trnh cho c nghim duy nht l x 2 . Nhn xt. y l mt phng trnh a thc thng thng, c nghim l x 2 nn vic phn tch thnh nhn t kh n gin; ci kh l bit nh gi phng trnh cn li v c nn tip tc tm cch gii n hay khng hay tm cch chng minh n v nghim. Trng hp bi cho phn tch thnh cc a thc khng c nghim n gin, bi ton tr nn kh khn hn rt nhiu; thm ch l ngay c vi nhng a thc bc bn. Chng hn nh khi gii phng trnh 2 x 4 3x 3 10 x 2 16 x 3 0 , nu tnh ton trn giy th khng phi d dng m c c phn tch (2 x 2 5 x 1)( x 2 x 3) 0 gii tng phng trnh tch. 2x 2 y 4 Bi 3. Gii h phng trnh 2x 5 2y 5 6 ( HSG B Ra Vng Tu) Li gii. iu kin: x, y 0 . Cng tng v hai phng trnh ca h, ta c: ( 2 x 5 2 x ) ( 2 y 5 2 y ) 10Tr phng trnh th hai cho phng trnh th nht, v theo v, ta c: 11 12. ( 2x 5 2x ) ( 2 y 5 2 y ) 2 5 2 2 2x 5 2x 2y 5 2yt a 2 x 5 2 x 0, b 2 y 5 2 y 0 . Ta c h sau:a b 10 b 10 a b 10 a a 5 5 5 5 5 2 b 5 50 20a 2a a b 2 a 10 a 2 Xt phng trnh 2 x 5 2 x 5 2 x 5 (5 2 x ) 2 2 x 5 25 2 x 10 2 x 2 x 2 x 2 .Tng t, ta cng c y 2 . Vy h phng trnh cho c nghim l ( x, y ) (2, 2) . Nhn xt. Ngoi cch gii tn dng tnh cht ca cc cn thc, ta cng c th t n ph ri bin i; trong phng trnh th hai, cc s hng t do c th khc nhau m li gii vn c tin hnh tng t. Chng hn, gii h phng trnh sau 2x 2 y 6 2x 5 2y 9 8 Bi 4. Gii h phng trnh sau 1 x x y 3 3 y 2 x y 1 8 y ( thi HSG Hi Phng, bng A) Li gii. iu kin y 0, x 1 0, x y 3 . y1 t a x , b x y 3, a, b 0 . H cho vit li l ya b 3 a 2, b 1 2 2 a 1, b 2 a b 5-Vi a 2, b 1 , ta c 12 13. 1 1 1 x 4 x 4 x 2, x y 3 1 x 4, x y 4 y y y 4 x 1 2 x 4 x 4 x 8 x 15 0, x 4 x 3, y 1 x 5, y 1 y 4 x y 4 x-Vi a 1, b 2 , ta c1 1 1 x 7 x 1 x 1, x y 3 2 x 1, x y 7 y y y 7 x x 4 10, y 3 10 x 2 8 x 6 0, x 7 x 4 10, y 3 10 y 7 x Th li, ta thy tt c u tha. Vy h phng trnh cho c 4 nghim l ( x, y ) (3,1), (5, 1), (4 10, 3 10), (4 10,3 10) .Nhn xt. Dng h phng trnh gii bng cch t n ph ny thng gp nhiu k thi, t H-C n thi HSG cp tnh v khu vc. Chng ta s cn thy n xut hin nhiu cc thi ca cc tnh c nu di y.4 x 2 y 4 4 xy 3 1 Bi 5. Gii h phng trnh 2 2 4 x 2 y 4 xy 2 ( thi HSG tnh Lm ng) Li gii. Ly phng trnh th nht tr phng trnh th hai, v theo v, ta c: y 4 2 y 2 4 xy 3 4 xy 1 0 ( y 2 1) 2 4 xy ( y 2 1) 0 ( y 2 1)( y 2 1 4 xy ) 0 y 1 y 1 y 2 1 4 xy 0-Nu y 1 , thay vo phng trnh u tin, ta c: 4 x 2 1 4 x 1 x( x 1) 0 x 0 x 1 . Th li, ta thy c hai nghim u tha mn. 13 14. -Nu y 1 , thay vo phng trnh u tin, ta c: 4 x 2 1 4 x 1 x ( x 1) 0 x 0 x 1 . Th li, ta thy c hai nghim u tha mn. -Nu y 2 1 4 xy 0 x 1 y2 (d thy trong trng hp ny y 0 ), thay vo phng trnh 4yu tin, ta c: 2 1 y2 1 y2 3 4 2 2 4 2 2 2 4 y 4 y 1 (1 y ) 4 y 4(1 y ) 4 ( y 1)(5 y 7) 0 . 4y 4y Suy ra y 1, x 0 v hai nghim ny nu trn. Vy h phng trnh cho c 4 nghim phn bit l ( x, y ) (1,1), (0,1), (1, 1), (0, 1) . Nhn xt. y l mt dng h phng trnh a thc kh kh, r rng nu phng trnh th hai, ngi ta chia hai v cho 2 th kh c th t nhn bit gi tr ny m nhn vo ri tr tng v nh trn. Vic pht hin ra gi tr 2 nhn vo c th dng cch t tham s ph ri la chn.x4 5 y 6 Bi 6. Gii h phng trnh trn tp s thc 2 2 x y 5x 6 ( thi chn i tuyn ng Nai) Li gii. Tr tng v hai phng trnh ca h, ta c x 4 x 2 y 2 5( y x) 0 ( x y ) x 2 ( x y ) 5 0 x y x 2 ( x y ) 5 -Nu x y , t phng trnh th nht ta c x 4 5 x 6 0 ( x 2 x 3)( x 2)( x 1) 0 x 2 x 1 , tng ng vi y 2 y 1 . Th li thy tha, ta c hai nghim ( x, y ) (2, 2), (1,1) . -Nu x 2 ( x y ) 5 y 5 x , thay vo phng trnh th nht ca h, ta c x2 5 x 4 5 2 x 6 x 6 5 x 3 6 x 2 25 0 x 14 15. ng thi, t h cho ta cng c 5 x 6 x 2 y 2 6 x 36 . 52216 96 312 6 6 Do 5 x 4 x 5. 4. 25 x 6 5 x 3 6 x 2 25 0 . 5 5 25 25 32Suy ra trong trng hp ny, h v nghim. Vy h cho c hai nghim l ( x, y ) (2, 2), (1,1) .3 2y 2 2 x y 1 x 1 Bi 7. Gii h phng trnh x2 y2 2x 4 y ( thi HSG H Tnh) Li gii. iu kin: xy 0, x 2 y 2 1 . t a x 2 y 2 1, b x , ab 0 . y2 3 2 3 1 b 2b 3 0 b 1, a 1 1 H cho tr thnh a b 2b 3 b b 3, a 9 a 2b 3 a 2b 3 a 2b 3 -Vi a 1, b 1 , ta c x 2 y 2 2, x y , ta tm c hai nghim l ( x, y ) (1, 1), (1,1) . -Vi a 9, b 3 , ta c x 2 y 2 10, x 3 y , ta tm c hai nghim l ( x, y ) (3,1), (3, 1) . Th li, ta u thy tha mn. Vy h cho c 4 nghim phn bit l ( x, y ) (1, 1), (1,1), (3,1), (3, 1) . Bi 8. Gii phng trnh3x 6 x2 7 x 1 ( thi chn i tuyn Lm ng)Li gii. iu kin x 1 . 15 16. Ta c( 3 x 6 2) ( x 2 4) ( x 1 1) 0 x2 x2 ( x 2)( x 2) 0 2 3 x 1 1 ( x 6) 2 3 x 6 4 1 1 ( x 2) x2 0 x 1 1 3 ( x 6) 2 2 3 x 6 4 x 2 1 1 x2 0 3 ( x 6)2 2 3 x 6 4 x 1 1 D thy phng trnh th hai v nghim v v tri lun dng nn phng trnh cho c nghim duy nht l x 2 . Nhn xt. Cch n gin hn dnh cho bi ny l chng minh hm ng bin, tuy nhin, cn ch xt x 1 trc khi o hm. x x y 1 1 Bi 9. Gii h phng trnh 2 2 y x 2y x y x 0 ( thi HSG tnh Qung Bnh) Li gii. iu kin x, x y 1 0 . Phng trnh th nht ca h tng ng vix x y 1 1 x x y 1 2 x y 1 1 y 2 x y 1 y 2 4( x y 1) ( y 2)2 4 x y 2 2 x Phng trnh th hai ca h tng ng vi y 2 x 2 y x y 2 x 0 ( y x )2 xy 2 y x y x y 1 x 1 y 2 2 x y 2 2 x y 2 2 x Ta c h mi l 4 2 2 y ( y 2) y ( y 2) y y 2 0 y x y x y 2 x 4 16 17. So snh vi iu kin ban u, ta thy c hai nghim trn u tha mn. 1 Vy h phng trnh cho c hai nghim l ( x, y ) ( , 1), (2, 4) . 4Bi 10. 1/ Gii bt phng trnh ( x 2 4 x) 2 x 2 3 x 2 0 . xy y 2 x 7 y 2/ Gii h phng trnh sau x 2 y x 12 ( thi HSG in Bin) Li gii. 1/ iu kin 2 x 2 3x 2 0 x 1 x 2 . Ta c 2x 4 x 0 x2 4x 0 ( x 4 x) 2 x 3 x 2 0 2 1 x x 2 2 x 3x 2 0 2 22Kt hp cc iu kin trn, ta c x 2 x 1 x 4. 2Vy bt phng trnh trn c nghim l x (,1 ] {2} [4, ) . 2x x y y 7 2/ iu kin y 0 . H cho tng ng vi ( x y ) x 12 y t u x y, v x , ta c h yu v 7 u 3, v 4 uv 12 u 4, v 3-Vi u 3, v 4 , ta c x y 4,x 3 x 3, y 1 , tha iu kin. y17 18. -Vi u 4, v 3 , ta c x y 3,12 3 x 4 x , y , tha iu kin. y 5 512 3 Vy h cho c hai nghim l ( x, y ) (3,1), ( , ) . 5 5 x 6 y 8 z10 1 . Bi 11. Gii h bt phng trnh 2007 2009 2011 x y z 1 ( thi chn i tuyn Bnh nh) Li gii. T bt phng trnh th nht ca h, ta c 1 x, y, z 1 . T hai bt phng trnh ca h, ta c x 2007 y 2009 z 2011 x 6 y 8 z10 x 6 (1 x 2001 ) y 8 (1 y 2001 ) z10 (1 z 2001 ) 0 T iu kin 1 x, y, z 1 , ta d dng thy rng x 6 (1 x 2001 ), y 8 (1 y 2001 ), z10 (1 z 2001 ) 0 . Do , phi c ng thc xy ra, tc l x 6 (1 x 2001 ) y 8 (1 y 2001 ) z10 (1 z 2001 ) 0 x, y , z 1 x, y , z 0 . Kt hp vi iu kin x 6 y 8 z10 1 , ta thy h bt phng trnh cho c cc nghim l( x, y , z ) (1, 0, 0), (0,1, 0), (0, 0,1) . Bi 12. 1/ Gii phng trnhx 1 1 x 2 x 1 3 xx2 x 2 y 2/ Gii h phng trnh 2 y y 2x ( thi HSG tnh Bn Tre) Li gii. 1/ iu kin x 1,3 x 0, x 1 3 x 1 x 3, x 1 . 18 19. Phng trnh cho tng ng vi 2 x 1 1 ( x 1) (3 x) x 1 3 xx 1 3 x ( x 1 3 x )( x 1 3 x ) x 1 3 x x 1 3 x 0 ( x 1 3 x ) 2 1 D thy phng trnh th nht v nghim nn ta ch xt ( x 1 3 x ) 2 1 ( x 1) (3 x ) 2 ( x 1)(3 x) 1 3 2 ( x 1)(3 x) 9 4( x 1)(3 x ) 4 x 2 8 x 3 0 x 2 7 2Vy phng trnh cho c hai nghim l x 2 7 . 22/ iu kin x, y 0 . D thy nu x 0 th y 0 v ngc li nn h c nghim ( x, y ) (0, 0) .t2 t 1 0, t 0 nn y l Ta xt x, y 0 . Xt hm s f (t ) , t 0 , ta thy f (t ) t 2 4 t hm ng bin. x f ( y) H cho c vit li l . Suy ra x y , thay vo h cho, ta c y f ( x) x 1 x x 2 x x x 1 2 x ( x 1)( x x 1) 0 x 3 5 2 2y 1 Tng ng vi hai gi tr ny, ta cng c y 3 5 2 Vy h cho c ba nghim l ( x, y ) (0, 0), (1,1), (3 5 3 5 , ). 2 2Nhn xt. Bi phng trnh th nht nu khng c bin i ph hp m t n ph th li gii s kh di dng v rc ri, chng ta cn ch tn dng nhng tnh cht ca cn thc, lng lin hp khai thc c im ring ca bi ton. 19 20. Bi 13. 1/ Gii phng trnh x 2 4 x 3 x 5 . 2/ Gii phng trnh x 3 x 2 3x 1 2 x 2 trn [2, 2] ( thi HSG tnh Long An) Li gii. 1/ iu kin x 5 . Phng trnh cho tng ng vi x 4 ( x 2 4 x 3)2 x 5 ( x 4)( x 3 4 x 2 6 x 1) 0 3 2 x 4x 6x 1 0Ta xt phng trnh x 3 4 x 2 6 x 1 0 (*) Hm s f ( x) x 3 4 x 2 6 x 1 c f ( x) 3 x 2 8 x 6 0 nn l ng bin; hn na,f (0). f (1) (1).2 0 nn phng trnh f ( x ) 0 c ng mt nghim thuc (0,1) . Ta s gii phng trnh (*) bng phng php Cardano. 4 2 61 0 . t y u v , ta c t x y , ta c (*) y 3 y 3 3 27 (u 3 v3 61 2 ) (3uv )(u v) 0 . 27 361 3 3 u v 27 Chn u v v sao cho . 2 uv 9 Gii h phng trnh ny, ta chn nghim u 31 2 (61 3 417), v . 54 9uT , ta tm c nghim ca phng trnh (*) lx x0 31 (61 3 417) 542 931 (61 3 417) 544 0.189464 320 21. Vy phng trnh cho c hai nghim l x 4, x x0 . 2/ iu kin x 2 . Phng trnh cho tng ng vi( x 3 x 2 3x 1)2 4( x 2) ( x 1)( x5 x 4 6 x3 2 x 2 9 x 7) 0 x 1 5 4 3 2 x x 6x 2 x 9x 7 0 Phng trnh x 5 x 4 6 x 3 2 x 2 9 x 7 0 c ng mt nghim thuc [2, 2] v n c gi tr gn ng l x x0 1.916086228 . Vy phng trnh cho c hai nghim phn bit l x 1, x x0 . Nhn xt. R rng phng trnh bc ba trn phi gii trc tip bng cng thc tng qut, iu ny t khi xut hin cc k thi HSG. i vi phng trnh th hai, vic xt x [2, 2] nu trong bi c th gi dng lng gic; tuy nhin, cch t x 2 cos cha c kt qu, mong cc bn tm hiu thm. Mt bi tng t xut hin trong k thi HSG BSCL nh sau Gii phng trnh 32 x 5 32 x 4 16 x3 16 x 2 2 x 1 0 . Phng trnh ny c gii bng cch t n ph y 2 x ri bnh phng ln, nhn vo hai v cho y 2 a v phng trnh quen thuc y 3 3 y y2.Bi ton nh th ny kh nh v phc tp! 1 y 2 x 2 y Bi 14. Gii h phng trnh sau x x 2 2 y ( x 1 1) 3 x 3( chn i tuyn trng Chuyn L Qu n, Bnh nh). Li gii. iu kin xc nh: x 0, y 0 . Phng trnh th nht ca h tng ng vi 1 xy 2 x 2 y x y 2 2 x x 2 xy y 2 y( x 2 x) 2 x x 0 x y 21 22. Xem y l phng trnh bc hai theo bin y, ta c x ( x 2 x)2 8 x x x 4 x x 4 x 2 ( x 2 x )2 0 . Do , phng trnh ny c hai nghim ly1 (2 x x ) ( x 2 x ) (2 x x ) ( x 2 x) x , y2 2 x, . 2 2Xt hai trng hp -Nu y x , thay vo phng trnh th hai ca h, ta c: x ( x 2 1 1) 3 x 2 3 . D thy: x ( x 2 1 1) 0 3 x 2 3 nn phng trnh ny v nghim. -Nu y 2 x , thay vo phng trnh th hai ca h, ta c: 2 x( x 2 1 1) 3 x 2 3 x 2 1.(2 x 3 ) 2 x 2xx2 1 (*)2x 33 3 khng tha mn ng thc nn ch xt x v php bin i trn l ph 2 2 2x hp). Xt hai hm s: f ( x ) x 2 1, x 0 v g ( x ) ,x 0. 2x 3 (d thy x Ta c: f ( x ) x 0 nn l hm ng bin, g ( x ) x2 1 Suy ra phng trnh (*) c khng qu mt nghim.2 3 (2 x 3 )2 0 nn l hm nghch bin.Nhm thy x 3 tha mn (*) nn y cng chnh l nghim duy nht ca (*).Vy h cho c nghim duy nht l ( x, y ) ( 3 , 2 3 ) .Nhn xt. Quan h ca x v y c che giu ngay trong phng trnh u tin, nu nhn thy iu th cc bc tip theo s rt d nhn bit. Bi ny tnh ton tuy rm r nhng hng gii rt r rng nn khng qu kh. 22 23. 2 x 2 y 3 xy 4 x 2 9 y Bi 15. Gii h phng trnh sau 2 7 y 6 2 x 9 x ( thi chn i tuyn Nha Trang, Khnh Ha) Li gii.4x2 2 x2 9 x 6 T phng trnh th nht, ta c y 2 , t phng trnh th hai, ta c y . 2 x 3x 9 7 Suy ra 4 x2 2x2 9x 6 28 x 2 (2 x 2 9 x 6)(2 x 2 3 x 9) 2 2 x 3x 9 7 ( x 2)(2 x 1)(2 x 2 9 x 27) 0 x 2 x -Nu x 2 , ta c y -Nu x 9 3 33 1 x 2 42 x 2 9 x 6 16 2 x 2 9 x 6 1 1 ; nu x , ta c y . 2 7 7 7 72x2 9x 6 9 3 33 vi 2 x 2 9 x 27 th y 3. 4 7Vy h phng trnh cho c bn nghim l ( x, y ) (2,16 1 1 9 3 33 ), ( , ), ( ,3) . 7 2 7 4Nhn xt. Bi ny c th cn nhiu bin i n gin hn nhng r rng cch rt y ra ri thay vo mt phng trnh nh trn l t nhin hn c. Bi 16. 1/ Gii phng trnh x 2 7 x 2 x 1 x 2 8 x 7 1 2 2 x y 3 2 x y 2/ Gii h phng trnh 3 x 6 1 y 4 ( thi HSG tnh Vnh Phc) Li gii. 1/ iu kin 1 x 7 . t a 7 x , b x 1, a, b 0 ab x 2 8 x 7 . 23 24. Phng trnh cho tr thnh b 2 2a 2b ab (a b)(b 2) 0 a b b 2 . -Nu a b th7 x x 1 7 x x 1 x 3 , tha iu kin bi.-Nu b 2 thx 1 2 x 3 .Vy phng trnh cho c nghim duy nht l x 3 . 2/ iu kin 2 x y 0, y 1 . Phng trnh th nht ca h tng ng vi (2 x y ) 2 2 x y 3 0 2 x y 1 2 x y 3 2 x y 1 y 1 2 x .Thay vo phng trnh th hai ca h, ta c3x 6 2x 4 .D thy v tri tng theo bin x nn phng trnh trn c khng qu mt nghim. Ta thy x 2 tha mn, suy ra x 2, y 3 . Vy h cho c nghim duy nht l ( x, y ) (2, 3) .Bi 17. Gii phng trnh sau x 4 2 x 3 2 x 2 2 x 1 ( x 3 x)1 x2 x ( thi HSG tnh H Tnh)Li gii. iu kin x (, 1] (0,1] . Nu x 1 th x 4 2 x 3 2 x 2 2 x 1 ( x 2 x) 2 ( x 1) 2 0, x 3 x x( x 2 1) 0 nn phng trnh trn khng c nghim tha x 1 . ng thi x 1 khng l nghim ca phng trnh nn ta ch xt x (0,1) . Phng trnh cho tng ng vi 2222( x 1) 2 x(1 x ) ( x 1) x (1 x ) t t x2 1 x (1 x 2 )2 x(1 x 2 ) 1 x2 1 x (1 x 2 ) x2 1 0 , phng trnh trn tr thnh t 2 1 t 2 t 2 0 t 2 (do t 0 ). tKhi 24 25. x2 1 2 2 ( x 2 1)2 4 x (1 x 2 ) x 4 2 x 2 1 4 x 4 x3 0x(1 x ) ( x 2 2 x 1)2 0 x 2 2 x 1 0 x 1 2 So snh vi iu kin nu, ta thy phng trnh trn c nghim duy nht l x 1 2 .Bi 18. Gii phng trnh 2 sin 2 x 3 2 sin x 2 cos x 5 0 . ( thi chn i tuyn trng THPT chuyn L Khit, Qung Ngi)Li gii. t a sin x, b cos x 1 a, b 1 . T phng trnh cho, ta c h sau: 4ab 3 2a 2b 5 0 2 2 a b 1 Ta c:4ab 3 2a 2b 5 0 4ab 3 2a 2b 5 0 (4ab 2 2a 2 2b 3) ( 2a 2b 2) 0 2(a b)2 2 2 (a b) 1 2 (a b 2 ) 0 ( 2a 2b 1)2 2 (a b 2 ) 0 Mt khc: a 2 b 2 1 nn a b 2(a 2 b 2 ) 2 a b 2 0 .ng thc xy ra khi v ch khi a b 2 . 2 2a 2b 1 0 ( 2a 2b 1)2 0 Do , t (*), suy ra: 2 2 (a b 2 ) 0 a b 2 D thy h ny v nghim. Vy phng trnh cho v nghim. 25 26. Nhn xt. y l dng phng trnh lng gic gii bng cch nh gi quen thuc. Ngoi cch t n ph a v i s hon ton nh trn, ta c th bin i trc tip trn phng trnh ban u, tuy nhin iu d lm chng ta lc sang cc hng thun ty lng gic hn v vic gii bi ton ny gp nhiu kh khn hn. Bi ny chnh l thi Olympic 30-4 nm 2000, lp 10 do trng L Hng Phong TP.HCM ngh. Li gii chnh thc cng ging nh trn nhng nguyn a sin x, b cos x . Bi 19. 1/ Gii phng trnh x 4 x 2 2 x 4 x 2 . 2 y ( x 2 y 2 ) 3x 2/ Gii h phng trnh 2 2 x( x y ) 10 y ( thi chn i tuyn THPT Chuyn Lam Sn, Thanh Ha) Li gii. 1/ iu kin 2 x 2 . Phng trnh cho tng ng vi( x 2) ( x 1) 4 x 2 ( x 2) 2 ( x 1) 2 (4 x 2 ) x( x 2)( x 2 2) 0 x 0 x 2 x 2 Th li ta thy tha. Vy phng trnh cho c 4 nghim l x 0, x 2, x 2 . 2/ Ta thy nu x 0 th y 0 v ngc li nn h phng trnh cho c nghim ( x, y ) (0, 0) . Xt trng hp xy 0 . Chia tng v phng trnh th nht cho phng trnh th hai, ta c 2 y( x 2 y 2 ) 3x 20 y 2 ( x 2 y 2 ) 3 x 2 ( x 2 y 2 ) 3 x 4 17 x 2 y 2 20 y 4 0 x( x 2 y 2 ) 10 y 5 ( x 2 4 y 2 )(3 x 2 5 y 2 ) 0 x 2 4 y 2 , x 2 y 2 3 2 3 2 y 3 x x 2 2 y.3 y 3 x 2 y x . -Nu x 2 4 y 2 , h cho tr thnh 4 2 xy 2 x.5 y 10 y 2 y 2 y 1 26 27. -Nu x 2 5 2 y , h cho tr thnh 315 2 2 2 y. y 3 x 3 3 x 2 4 135 3 4 y 9 x 4 y 9 x . 4 4 8 2 16 y 135 135 4 xy 15 x. y 10 y y 3 2 Vy h cho c 5 nghim l ( x, y ) (0, 0),(2,1), ( 2, 1), (Bi 20. Gii phng trnh:34 4 15 135 15 135 , ), ( 4 , ). 4 2 2 2 135 2 135x 6 x2 7 x 1 . ( thi HSG tnh Lm ng)Li gii. iu kin x 1 . D thy x 1 khng l nghim ca phng trnh nn ta ch xt x 1 . Ta c3x 6 x 2 7 x 1 3 x 6 x 2 x 1 7 (*)Xt hm s f (t ) 3 t 6 t 2 t 1, t 1 f (t ) 1 3 2 (t 6)2 2t 1 0, t 1 . Do 2 t 1hm ny ng bin. T suy ra phng trnh (*) trn c khng qu mt nghim; mt khc f (2) 7 nn phng trnh cho c nghim duy nht l x 2 . 5( x y ) 6( x z ) x y 6 xy x z 5 xz 4 6( z y ) 4( x y ) 5 Bi 21. Gii h phng trnh z y 4 zy x y 6 xy 4( x z ) 5( y z ) 6 x z 5 xz y z 4 yz ( chn i tuyn trng PTNK, TPHCM) Li gii. t a x y yz zx ,b ,c . H cho tr thnh x y 6 xy y z 4 yz z x 5 zx 27 28. 1 4 5a a 8 c 5a 6c 4 6 3 b 6b 4a 5 4a 6b 5 4 4 5a 4c 5b 6 9 5b 4 6 c 16 6 1 x y 1 1 6 1 33 14 x y 6 xy 8 x y 7 x 14 x 33 7( x y ) 6 xy 3 14 yz 1 1 1 45 y z 12 yz 12 y Do 45 y z 4 yz 4 7( z x) 45 zx y z y 14 14 zx 1 1 45 1 123 9 z 124 z 14 z x 5 zx 16 z x 7 14 14 14 , , ). 33 45 123 Nhn xt. Bi ny c hnh thc kh phc tp v cc h s xem ra rt khc nhau; tuy nhin, nu quan st k, chng ta s d dng tm ra cc n ph cn thit lm n gin ha bi ton.Vy h cho c nghim l ( x, y , z ) (Bi 22. 1/ Gii phng trnhx 2 y 1 3 z 2 1 ( x y z 11) 2x 2 121 x 2x 27 2 2/ Gii h phng trnh 9 x 2 y 2 xy 3x 4 y 4 0 ( thi HSG tnh Qung Nam) Li gii. 1/ iu kin x 0, y 1, z 2 . Phng trnh cho tng ng vi2 x 4 y 1 6 z 2 x y z 11 ( x 1) ( y 1 2) 2 ( z 2 3) 2 0 x 1 y 1 2 z 2 3 0 x 1, y 5, z 11Vy phng trnh cho c nghim l ( x, y , z ) (1,5,11) .28 29. 2/ Phng trnh th hai ca h tng ng vi y 2 ( x 4) y x 2 3x 4 0 . y l phng trnh bc hai theo bin y nn cn c iu kin 4 ( x 4) 2 4( x 2 3x 4) 3 x 2 4 x 0 0 x . 3 x 2 4 4 16 24 121 Do x 2 2 x 27 2 ( )2 2. 27 3 9 3 3 9 9 9T bt ng thc trn v phng trnh th nht ca h, ta suy ra x 4 . 34 4 4 8 16 4 4 Do y 2 ( 4) y ( ) 2 3.( ) 4 0 y 2 y 0 ( y ) 2 0 y 3 3 3 3 9 3 3Vy h cho c nghim l x y 4 . 3Bi 23.x 2 2ax b 1/ Tm tt c cc gi tr ca a, b phng trnh 2 m c hai nghim phn bit vi bx 2ax 1 mi tham s m. y xy 2 6 x 2 2/ Gii h phng trnh 3 3 3 1 x y 19 x ( thi HSG vng tnh Bnh Phc) Li gii. 1/Trc ht, ta s tm nghim chung, nu c, ca hai phng trnh bc hai sau:x 2 2ax b 0 v bx 2 2ax 1 0 . Gi s x0 l nghim chung , ta c: 2 2 x0 2ax0 b 0 v bx0 2ax0 1 0 . Tr tng v hai phng trnh ny, ta c: 2 (1 b)( x0 1) 0 b 1 x0 1 .x 2 2ax 1 m 1 m, x 2 2ax 1 0 . D thy x 2 2ax 1 nu m 1 th phng trnh ny v nghim, nu m 1 th phng trnh ny c v s nghim, khng tha mn bi.-Nu b 1 th phng trnh cho tr thnh29 30. -Nu b 1 th x0 1 , tng ng vi 1 2a b 0 hoc 1 2a b 0 . Do , khi 1 2a b 0 hoc 1 2a b 0 th tng ng hai phng trnh cho c nghim chung l x0 1 v x0 1 . Phng trnh ban u tng ng vix 2 2ax b x 2 2ax b m(bx 2 2ax 1) m 0 bx 2 2ax 1 bx 2 2ax 1hay (1 bm) x 2 2(a am) x b m 0 (*) v bx 2 2ax 1 0 . Ta thy rng phng trnh (*) c khng qu hai nghim nn mun phng trnh cho c hai nghim phn bit vi mi m th hai phng trnh x 2 2ax b 0 v bx 2 2ax 1 0 khng c nghim chung, ng thi phng trnh (*) phi c ng hai nghim phn bit, tc l1 2a b 0,1 2a b 0 1 bm 0, m (a am)2 (1 bm)(b m) 0, m T iu kin th hai, ta thy b 0 , khi , h iu kin trn tr thnh1 1 1 2a 0,1 2a 0 a a 2 2 2 (a am) m 0, m a 2 m 2 (2a 2 1)m a 2 0, m (2a 2 1)2 4a 4 0, a 0 1 1 1 a , a 0 a a 2 2 2 4 a 2 1 0 Vy cc gi tr a, b tha mn bi l a 1 1 a v b 0 . 2 2 y xy 2 6 x 2 2/ Gii h phng trnh 3 3 3 1 x y 19 x Ta thy y 0 khng l nghim ca h phng trnh nn ta ch xt y 0 , ta c bin i saux 2 x 2 1 1 y x 6( y ) y x 6( y ) 1 x3 19( x )3 ( x 1 )3 3 x ( x 1 ) 19( x )3 3 y y y y y y 30 31. x 1 x 6( )2 vo phng trnh th hai, ta c y y x x x x x x x 1 216( )6 18( )3 19( )3 216( )6 ( )3 0 . y y y y y y y 6Thay-Nux 0 x 0 , thay vo h cho, ta thy khng tha mn. y-Nu y 6 x , thay vo phng trnh th nht ca h, ta c 1 1 6 x 36 x3 6 x 2 6 x3 x 2 x 0 x(3x 1)(2 x 1) 0 x x . 3 2 1 1 Vi x , ta c y 6 x 2 ; vi x , ta c y 6 x 3 . 3 2Th li u thy tha. 1 1 Vy h cho c hai nghim phn bit l ( x, y ) ( , 2), ( ,3) . 3 2Nhn xt. bi 1, bc tm nghim chung ca hai phng trnh lm n gin ha vic xt iu kin ca nghim xem c tha mn phng trnh hay khng, v r rng f ( x) mg ( x) 0 f ( x) f ( x ) mg ( x) nn nu x l nghim ca phng trnh m 0 g ( x) g ( x) g ( x) 0 cho m li khng tha mn iu kin xc nh ca mu th n l nghim chung ca f ( x ), g ( x ) ( y l xt vi mi m nn c c nhng gi tr m khc 0). Bi 2 xut hin kh nhiu trong cc ti liu luyn thi H v vic tm ra cch chia nh th cng kh l m mn, chng ta c th rt y t phng trnh di thay ln ri nh gi phng trnh mt n x thu c. Bi 24. x 2 y 2 z 2 2010 2 1/ Gii h phng trnh: 3 3 3 3 x y z 2010 2/ Gii phng trnh: 32 x3 x2 3x32x x3 3x 2 0 ( thi chn i tuyn Ninh Bnh)Li gii. 1/ T phng trnh th nht ca h, ta c x , y , z 2010 . 31 32. Suy ra x 3 y 3 z 3 x 3 y 3 z 3 2010( x 2 y 2 z 2 ) 20103 . T phng trnh th hai suy ra ng thc phi xy ra, tc l x 2 (2010 x) 0 x 0 x 2010 2 y (2010 y ) 0 y 0 y 2010 2 z 0 z 2010 z (2010 z ) 0Kt hp vi phng trnh th nht, ta thy h cho c ba nghim phn bit l( x, y , z ) (2010, 0, 0), (0, 2010, 0), (0, 0, 2010) .2/ Phng trnh cho tng ng vi 32 x3 x2 2 x3 x 2 3 x32x x3 2 xXt hm s f (t ) 3t t , t , ta c f (t ) 3t .ln 3 1 0, t nn y l hm ng bin. Phng trnh trn chnh lf (2 x 3 x 2) f ( x 3 2 x) 2 x 3 x 2 x3 2 x x3 3x 2 0 ( x 2)( x 1) 2 0 x 2 x 1Vy phng trnh cho c hai nghim l x 2, x 1 . Bi 25. 2 x y x 1 2( x 1) 2(2 x y ) 2 1/ Gii bt phng trnh sau 2 y 4 x x 1 17 0 2/ Vi n l s nguyn dng, gii phng trnh1 1 1 1 ... 0. sin 2 x sin 4 x sin 8 x sin 2 n x( thi HSG tnh Khnh Ha) Li gii. 1/ iu kin x 1 . Bnh phng hai v ca bt phng trnh th nht ca h, ta c (2 x y )2 x 1 2(2 x y ) x 1 2( x 1) 2(2 x y ) 2 (2 x y )2 x 1 2(2 x y ) x 1 0 (2 x y x 1) 2 0 2 x y x 1 0 y 2 x x 1 32 33. Thay vo phng trnh th hai ca h, ta c(2 x x 1)2 4 x x 1 17 0 4 x ( x 1) 4 x x 1 4 x x 1 17 0 9 4 x 2 x 18 0 x 2 x 4 Ta thy ch c x 2 l tha mn, khi , tng ng ta c y 3 . Vy h bt phng trnh cho c nghim duy nht l ( x, y ) (2,3) . 2/ iu kin 2i x k 2 , i 1, 2,3,..., n; k x k , i 0,1, 2,..., n 1 . 2iTrc tin, ta s rt gn v tri ca phng trnh cho. Ta c bin i sau cot a cot 2a cos a cos 2a 2cos2 a cos 2a 1 . sin a sin 2a sin 2a sin 2aDo 1 1 1 1 ... 0 sin 2 x sin 4 x sin 8 x sin 2n x n n 1 0 (cot 2i 1 x cot 2i x) 0 cot 2n x cot x i i 1 sin 2 x i 1 k 2 n x x k x n , k 2 1D thy nghim ny tha mn iu kin ban u. Vy phng trnh cho c nghim l x k ,k . 2n 1Bi 26. 1/ Gii phng trnh sau2/ Gii phng trnh log 33 sin 2 x cos 2 x 5sin x (2 3) cos x 3 3 1. 2 cos x 32x 1 3x2 8x 5 ( x 1) 2 ( thi HSG tnh Thi Bnh) 33 34. Li gii. 1/ iu kin cos x 3 5 x k 2 . 2 6Phng trnh cho tng ng vi3 sin 2 x cos 2 x 5sin x (2 3) cos x 3 3 2 cos x 3 3 sin 2 x cos 2 x 5sin x 3 cos x 3 0 2 3 sin x.cos x 1 2sin 2 x 5sin x 3 cos x 3 0 2sin 2 x sin x(2 3 cos x 5) 3 cos x 2 0 t t sin x, t 1 . Ta c 2t 2 t (2 3 cos x 5) 3 cos x 2 0 (*) y l phng trnh bc hai bin t c (2 3 cos x 5) 2 8( 3 cos x 2) 12 cos 2 x 12 3 cos x 9 (2 3 cos x 3) 2Do , phng trnh (*) c hai nghim lt(2 3 cos x 5) (2 3 cos x 3) 1 (2 3 cos x 5) (2 3 cos x 3) t 3 cos x 2 4 2 4-Nu t 1 1 7 sin x x k 2 x k 2 , k (tha mn). 2 2 6 6 -Nu t 3 cos x 2 sin x 3 cos x 2 sin( x ) 1 x k 2 , k (tha mn). 3 6 Vy phng trnh cho c ba h nghim l x 7 k 2 , x k 2 , x k 2 , k . 6 6 61 2/ iu kin 2 x 1 0, x 1 0 x , x 1 . 2Phng trnh cho tng ng vi 2 x 1 log 3 3 x 2 8 x 4 log 3 (2 x 1) log 3 3( x 1) 2 3( x 1) 2 (2 x 1) 2 3( x 1) log 3 (2 x 1) (2 x 1) log3 3( x 1) 2 3( x 1) 234 35. Xt hm s f (t ) log 3 t t , t 0 , ta c f (t ) 1 1 0, t 0 nn y l hm ng bin. t ln 3Phng trnh trn chnh l f (2 x 1) f (3( x 1)2 ) 2 x 1 3( x 1) 2 3 x 2 8 x 4 0 x 2 x2. 32 Ta thy hai nghim ny u tha mn nn phng trnh cho c hai nghim l x , x 2 . 3Nhn xt. bi phng trnh lng gic, n lc rt gn c thnh mt phng trnh ch cha sin x, cos x ; ta thng dng cch t n ph nh trn i s ha vic gii bi ton, khng phi d dng c th tm ra cch phn tch nhn t nh trn, nht l nhng bi ton di dng hn. Nu t t sin x khng thnh cng, ta hon ton c th chuyn sang t cos x th v chng hn, nh bi ton trn, nu t t cos x th li gii s khng cn d dng na. Trong thi H khi B nm 2010 cng c mt bi tng t, gii phng trnh sau sin 2 x cos 2 x 3sin x cos x 1 0 . Bng vic p dng cng thc nhn i a phng trnh v dng f (sin x, cos x ) 0 , ta tin hnh t n ph t sin x phn tch thnh nhn t, li gii kh r rng v t nhin.Cc bn th gii thm bi ton sau 4sin 3 x sin x.cos x(7 sin x 3cos x ) (sin 2 x cos 2 x) 5(sin x cos x) 2 cos 2 x 0 . Bi 27. x 2 1 y 2 xy y 1/ Gii h phng trnh . y x y2 1 x2 2/ Gii phng trnh lng gic2 2 2 2 sin 2 x tan x cot 2 x ( thi HSG tnh Ph Th)Li gii. 1/ Ta thy h phng trnh ny khng c nghim tha y 0 nn ta ch xt y 0 , khi ,x2 1 x y 1. phng trnh th nht ca h tng ng vi y 35 36. x2 1 , v x y . Ta c h t u ya b 4 b 4 a b 4 a b 4 a b 3 1 1 2 2 a 1 2a a 1 (a 1) 0 b 2 a 2 a a x 1 x2 1 1 x2 1 3 x x2 x 2 0 y 2 Ta c y x 2 y 3 x y 3 x x y 3 y 5 Vy h phng trnh cho c hai nghim l ( x, y ) (1, 2), (2,5) . 2/ iu kin cos x 0,sin 2 x 0, tan x cot 2 x 0 sin x.cos x 0,sin x cos 2 x 0 cos x sin 2 x2 sin 2 x 1 2 sin 2 x sin x.cos x 0, 0 sin x cos x 0 sin 2 x x (k 2 , k 2 ) ( k 2 , k 2 ), k 2 2Ta bin i phng trnh cho tng ng vi2 1 1 2 sin 2 x ( 2 1) sin 2 x 1 2 sin 2 x 1 sin 2 x t t sin 2 x , 0 t 1 . Phng trnh trn chnh l( 2 1)t 1 2t 2 (t 1)( 2t 1) 0 t 1 t -Nu t 1 sin 2 x 1 x -Nu t 1 (tha iu kin). 2 k , k . 41 1 5 sin 2 x x k k , k . 2 12 12 2So snh vi iu kin ban u, ta thy phng trnh cho c ba h nghim l 36 37. x 5 k 2 , x k 2 , x k 2 , k . 4 12 12Bi 28. Gii phng trnh: 24 x 2 60 x 36 1 1 0 5x 7 x 1 ( thi HSG tnh Qung Ninh)Li gii. iu kin: x f (t ) 2t 1 7 , t 1 . Ta c: . Xt hm s f (t ) t 2 5 t 11 0, t 1 nn hm ny ng bin. 2(t 1) t 17 7 1 v 5 x 6 5. 6 1 nn phng trnh cho tng ng vi 5 5 1 1 3 (5 x 6)2 x2 f (5 x 6) f ( x) 5 x 6 x x . 2 5x 7 x 1Do x Th li ta thy tha iu kin bi. Vy phng trnh cho c nghim duy nht l x Bi 29. Gii phng trnh3 . 23 x 3 2 x 2 2 3 x 3 x 2 2 x 1 2 x 2 2 x 2 ( thi chn i tuyn trng THPT Chuyn HSP H Ni)Li gii. 2 3 3 x 2 x 2 0 iu kin xc nh 3 2 3x x 2 x 1 0 Theo bt ng thc AM GM th3 x 3 2 x 2 2 1. 3 x 3 2 x 2 2 ng thc xy ra khi v ch khi1 (3 x3 2 x 2 2) 3x 3 2 x 2 3 2 23 x 3 2 x 2 2 1 x 1 .37 38. 1 (3 x3 x 2 2 x 1) 3 x3 x 2 2 x 3 x x 2 x 1 1. 3 x x 2 x 1 2 2 3232ng thc xy ra khi v ch khi3x 3 x 2 2 x 1 1 x 1 .3 x 3 2 x 2 3 3 x 3 x 2 2 x 3 x 2 2 x 3 3 x 2 x 2 3 x x 2 x 1 2 2 2 32323x 2 2 x 3 (3x 2 2 x 3) ( x 1)2 2x2 2 x 2 2 2 ng thc xy ra khi v ch khi ( x 1)2 0 x 1 . Do , ta lun c3 x 3 2 x 2 2 3 x 3 x 2 2 x 1 2 x 2 2 x 2 .ng thc phi xy ra, tc l x 1 . Th li thy tha. Vy phng trnh cho c nghim duy nht l x 1 .Nhn xt. Bi ny khng qu kh v ch p dng cc nh gi rt quen thuc ca BT. Tuy nhin, xc nh c hng i ny cng khng phi n gin; thng thng sau khi nhm ra c nghim l x 1 v ng trc mt phng trnh v t c cha cn th ny, ta hay dng cch nhn lng lin hp; th nhng, cch ri cng s i vo b tc cng nhng tnh ton phc tp.(2 x 2 3 x 4)(2 y 2 3 y 4) 18 Bi 30. Gii h phng trnh 2 2 x y xy 7 x 6 y 14 0 ( thi chn i tuyn trng THPT Chuyn HSP H Ni)Li gii. Xt ng thc x 2 y 2 xy 7 x 6 y 14 0 (*) Ta xem (*) l phng trnh bc hai theo bin x, vit li l x 2 x( y 7) y 2 6 y 14 0 .38 39. Phng trnh ny c nghim khi y ( y 7)2 4( y 2 6 y 14) 0 3 y 2 10 y 7 0 1 y 7 . 3Hon ton tng t, xem (*) l phng trnh bc hai theo bin y, vit li l y 2 y ( x 6) ( x 2 7 x 14) 0 .Phng trnh ny c nghim khi x ( x 6)2 4( x 2 7 x 14) 0 3 x 2 16 x 20 0 2 x Ta xt hm s f (t ) 2t 2 3t 4, t f (t ) 4t 3 0 t 10 . 33 1. 4Suy ra, trn [1, ) , hm s ny ng bin. Ta cf ( x ) f (2) 6, f ( y ) f (1) 3 f ( x). f ( y ) 3.6 18 . T phng trnh th nht ca h th ta thy ng thc phi xy ra, tc l x 2, y 1 . Thay hai gi tr ny vo (*), ta thy khng tha. Vy h phng trnh cho v nghim. Nhn xt. tng gii ca bi ny khng kh v cng kh quen thuc khi ch cn tm min xc nh ca bin thng qua vic tnh Delta ca mt phng trnh bc hai; tuy trong li gii trn c kho st hm s nhng thc ra cc kt qu c th chng minh bng bt ng thc i s thun ty nn cng c gii chnh ca bi ny l i s. V do vic hai biu thc ca x v y phng trnh u ca h ging nhau c th dn n nh gi sai hng m dng gii tch, xt hm s khai thc phng trnh u tin trong khi iu khng em li kt qu g. Cc h s c chn ra s rt p chnh l u im ni bt ca bi ton ny.2(2 x 1)3 2 x 1 (2 y 3) y 2 Bi 31. Gii h phng trnh 4x 2 2y 4 6 ( thi chn i tuyn trng THPT chuyn Lng Th Vinh, ng Nai)Li gii. 39 40. iu kin xc nh: x 1 , y 2. 2Xt hm s: f (t ) 2t 3 t , t (0; ) . Suy ra: f (t ) 6t 2 1 0 nn y l hm ng bin. T phng trnh th nht ca h, ta c: f (2 x 1) f ( y 2 ) 2 x 1 y 2 . Thay vo phng trnh th hai, ta c: 44 y 8 2 y 4 6 (*)Ta thy hm s: g ( y ) 4 4 y 8 2 y 4 6, y (2, ) c o hm l:1g ( y ) 4(4 y 8)31 0, y (2, ) nn ng bin.2y 4Hn na: g (6) 4 4.6 8 2.6 4 6 0 nn (*) c ng mt nghim l y 6 . Vi y 6 , ta c x 1 . 21 Vy h cho c nghim duy nht l ( x, y ) ( , 6) . 2 Nhn xt. Dng ton ng dng trc tip tnh n iu vo bi ton n gin ha biu thc rt thng gp. Hng gii ny c th d dng pht hin ra t phng trnh th nht ca h, x v y nm v mi v ca phng trnh v quan st k hn s thy s tng ng ca cc biu thc v dn n xt mt hm s nh nu trn. tng bi ny hon ton ging vi bi 5 thi H mn ton khi A nm 2010. (4 x 2 1) x ( y 3) 5 2 y 0 Gii h phng trnh . 2 2 4 x y 2 3 4 x 7 x 4 x3 y 9 y y 3 x x2 y 2 9 x Bi 32. Gii h phng trnh 3 3 x( y x ) 7 40 41. ( thi chn HSG tnh Hng Yn) Li gii. Ta c x 4 x 3 y 9 y y 3 x x 2 y 2 9 x ( x 4 xy 3 ) ( x 3 y x 2 y 2 ) 9( x y ) 0 ( x y ) x( x 2 xy y 2 ) x 2 y 9 0 ( x y ) x( x y )2 9 0 T phng trnh th hai ca h, ta thy x y nn t bin i trn, suy ra: x( x y )2 9 0 x( x y )2 9 (*)Ta c: x( y 3 x 3 ) 7 y 3 x 3 7 7 y 3 x3 . x x7 Thay vo (*), ta c: x( x 3 x 3 )2 9 . x Ta s chng minh rng v tri l mt hm ng bin theo bin x. Tht vy: 2 7 2 x 2 2 x 3 x3 7 3 x3 7 x( x x ) x x x x 332 7 7 x 2 x . x x 3 x3 x3 2 x 3 x6 7 x2 3 x x 4 7 x x 32 332T (*) suy ra x 0 v trong biu thc trn, cc s m ca bin x u dng nn y l hm ng bin; suy ra n c khng qu mt nghim. Thay trc tip x 1 vo biu thc, ta thy tha. Vy h cho c ng mt nghim l: ( x, y ) (1, 2) . Nhn xt. im c bit ca bi ny l x l c h phng trnh mi sau khi bin i, nu nh ta dng cch i s trc tip, phn tch ra c mt nghim x = 1 th phng trnh bc cao cn li kh m gii c. Cch lp lun theo tnh n iu ca hm s th ny va trnh c iu va lm cho li gii nh nhng hn. 2 y 3 2 x 1 x 3 1 x y Bi 33. Gii h phng trnh 2 y 2 x 1 2 xy 1 x 41 42. ( thi chn i tuyn chuyn Nguyn Du, k Lk) Li gii. iu kin 1 x 1 . t a 1 x 0 x 1 a 2 . Phng trnh th nht ca h tng ng vi 2 y 3 2(1 a 2 )a 3a y 2 y 3 y 2a 3 a D thy hm s f (t ) 2t 3 t , t c f (t ) 6t 2 1 0, t nn ng thc trn c vit li l f ( y ) f (a ) y a y 1 x . Thay vo phng trnh th hai ca h, ta c 1 x 2 x2 1 2 x 1 x2t x cos t , t 0, , phng trnh trn tr thnh1 cos t 2 cos 2 t 1 2cos t 1 cos 2 t t t cos 2t sin 2t 2 sin 2 sin(2t ) 2 2 4 4 t 2t 4 2 k .2 t 6 k 3 t sin sin(2t ) ,k ,k t 2 4 3 4 2t k .2 t k 10 4 2 5 2sin 2Do t 0, nn t hai h phng trnh trn, ta ch nhn gi tr t Khi x cos3 . 103 3 3 , y 1 cos 2 sin . 10 10 20Vy h cho c nghim duy nht l ( x, y ) (cos3 3 , 2 sin ) . 10 20 x 3 y 3 35 Bi 34. Gii h phng trnh: 2 2 2 x 3 y 4 x 9 y ( thi HSG tnh Yn Bi) 42 43. Li gii. Phng trnh th hai ca h tng ng vi (6 x 2 12 x 8) (9 y 2 12 y 27) 35 Thay vo phng trnh th nht ca h, ta c: x 3 y 3 (6 x 2 12 x 8) (9 y 2 12 y 27) ( x 2) 3 ( y 3) 3 x y 5 Li thay vo phng trnh th hai ca h, ta c: 2( y 5)2 3 y 2 4( y 5) 9 y 5 y 2 25 y 30 0 ( y 2)( y 3) 0 y 2 y 3 . Vi y 2 , ta c x 3 , vi y 3 , ta c x 2 . Th li ta thy tha. Vy h phng trnh cho c hai nghim l ( x, y ) (2, 3), (3, 2) . Nhn xt. Dng ton da trn hng ng thc ny xut hin kh nhiu, chng hn trong thi VMO 2010 va qua; nu chng ta thy cc biu thc ca x v y trong h phng trnh cha y cc bc th kh nng gii theo cch dng hng ng thc l rt cao. x3 y3 9 Mt bi ton tng t, gii h phng trnh sau 2 . 2 x 2 y x 4 y Bi 35. Gii phng trnh 2 3 2 x 1 27 x 3 27 x 2 13x 2 ( thi HSG Hi Phng, bng A1) Li gii. Phng trnh cho tng ng vi (3 x 1)3 2(3 x 1) (2 x 1) 2 3 2 x 1 . Xt hm s f (t ) t 3 2t , t . Ta thy f (t ) 6t 2 1 0, t nn y l hm ng bin. Phng trnh trn chnh l f (3 x 1) f ( 3 2 x 1) 3 x 1 3 2 x 1 (3x 1)3 2 x 1 27 x 3 27 x 2 7 x 0 x (27 x 2 27 x 7) 0 x 0Vy phng trnh cho c nghim duy nht l x 0 . 43 44. 1 1 2 2 x 2 y 2( x y ) Bi 36. Gii h phng trnh 1 1 y2 x2 x 2y ( thi chn i tuyn Qung Ninh) Li gii. iu kin x, y 0 . Cng tng v ca hai phng trnh, ta c2 x 2 3 y 2 2 x 3 3xy 2 xLy phng trnh th nht tr phng trnh th hai, v theo v, ta c 1 3x 2 y 2 1 3x 2 y y 3 y Do 3 3 1 x 2 x 3 3xy 2 3 x3 3 xy 2 y 3 3 x 2 y 3 ( x y )3 x y 3 3 2 3 2 3 2 3 2 3 3 x y 1 1 y 3 x y 1 x 3 xy y 3 x y 1 ( x y ) y 3 1 2Th li, ta thy nghim ny tha. 3Vy h phng trnh cho c nghim l x 3 3 1 3 1 ,y . 2 2Nhn xt. Dng ton ny cng xut pht t vic khai trin cc hng ng thc, nhng y l da trn tnh i xng tm ra s bt i xng nhm sng to ton th v. Cch gii bi ny theo hng trn l quen thuc v tt hn c, mt bi tng t trong thi HSG ca TPHCM l1 1 4 4 x 2 y 2( y x ) Gii h phng trnh 1 1 (3x 2 y 2 )( x 2 3 y 2 ) x 2y Bi ny to ra t khai trin nh thc Newton bc nm, nu ta xt khai trin bc by th bi ton thu c s rt n tng.Bi 37. Gii h phng trnh 44 45. x 3 3 x 12 y 50 3 y 12 y 3 z 2 z 3 27 x 27 z ( thi chn i tuyn trng THPT Phan Chu Trinh, Nng) Li gii. Ta c x 3 3 x 12 y 50 48 12 y x3 3x 2 12(4 y ) ( x 2)( x 1)2(1)y 3 12 y 3 z 2 3z 18 y 3 12 y 16 3( z 6) ( y 4)( y 2)2(2)z 3 27 x 27 z 27 x 54 z 3 27 z 54 27( x 2) ( z 6)( z 3)2(3)-Nu x 1 th ( x 2)( x 1)2 0 , t (1) suy ra y 4 hay ( y 4)( y 2)2 0 , t (2) suy ra z 6 hay ( z 6)( z 3)2 0 , t (3) suy ra x 2 , mu thun.Do , x 1 khng tha mn h, ta ch xt x 1 ( x 1)2 0 . Chng minh hon ton tng t, ta cng c: ( y 2)2 0,( z 3)2 0 . T (2) suy ra y 4, z 6 cng du. T (3) suy ra x 2, z 6 cng du. T , ta c: x 2, y 4 cng du. Hn na, t (1), ta thy x 2, ( y 4) cng du, tc l: 0 ( x 2)( y 4) 0 . Do : x 2 hoc y 4 . T cc phng trnh (1), (2), (3), d thy c hai trng hp trn u cho ta kt qu l:x 2, y 4, z 6 . Vy h cho c nghim duy nht l ( x, y, z ) (2, 4, 6) . Nhn xt. Mu cht ca bi ton l phi c c cc phn tch (1), (2), (3) trn. iu ny ch c th thc hin c khi on c nghim ca bi ton l x 2, y 4, z 6 v c th tc 45 46. gi bi ton cng xut pht t cc ng thc bin i c nh trn. Dng ny cng tng xut hin trong thi HSG ca TPHCM nm 2006 2007 vi cch gii tng t. x3 y 3x 4 Gii h phng trnh 2 y 3 z 6 y 6 3 z 3 x 9 z 8 Bi 38. Gii phng trnh3x9 9 x 2 1 2x 1 3 ( thi chn i tuyn Ph Yn)Li gii. Phng trnh cho tng ng vi x 9 9 x 2 1 3(2 x 1)3 x 9 9 x 2 1 24 x 3 36 x 2 18 x 3 x 9 3x 3 (27 x 3 27 x 2 9 x 1) 9 x 3 x 9 3x 3 (3 x 1)3 3(3x 1)Xt hm s f (t ) t 3 3t , t , ta c f (t ) 3t 2 3 0, t nn y l hm ng bin. Phng trnh trn c vit li l f ( x 3 ) f (3x 1) x 3 3x 1 . (*) Trc ht, ta xt cc nghim tha mn 2 x 2 ca (*). t x 2 cos , [0, ] , khi (*) 8 cos3 6 cos 1 2 cos 3 1 cos 3 cos2 x k . 3 9 3 5 7 , , , tng ng, ta c 9 9 9 5 7 cc nghim ca (*) l x 2 cos , x 2 cos , x 2 cos . R rng ba nghim ny l phn 9 9 9 bit v (*) l phng trnh bc ba, c khng qu ba nghim nn y cng chnh l tt c cc nghim ca n.M [0, ] nn ta ch chn 3 nghim ca h trn l Vy phng trnh cho c cc nghim l x 2 cos 5 7 , x 2 cos , x 2 cos . 9 9 9Bi 39.46 47. 1/ Gii phng trnh sau x 1 x 1 2 x x 2 2 y 3 y x 3 3x 2 4 x 2 2/ Gii h phng trnh sau 2 1 x y 2 y 1 ( thi HSG tnh Ngh An) Li gii. 1/ iu kin 1 x 2 . t t x 1 2 x 0 , ta ct 2 3 2 x2 x 2 (t2 3 2 ) x2 x 2 . 2Phng trnh cho tng ng vi ( x 1 2 x ) ( x 2 x 2) 3 2 t (t2 3 2 ) 3 2 2 t 4 6t 2 4t (3 4 2) 0 t 4 (1 2)t 3 (1 2)t 3 (3 2 2)t 2 (2 2 3)t 2 (1 2)t (5 2)t (3 4 2) 0 (t 2 1)[t 3 (1 2)t 2 (2 2 3)t (5 2)] 0 t 2 1 t 3 (1 2)t 2 (2 2 3)t (5 2) 0 D thy phng trnh th hai khng c nghim dng nn ta ch xt t 2 1 . Khi x 1 2 x 2 1 3 2 x 2 x 2 3 2 2 x 2 x 0 x 0 x 1 , tha.Vy phng trnh cho c hai nghim l x 0, x 1 .2/ iu kin 1 x 1, 0 y 2 . Phng trnh th nht ca h tng ng vi y 3 y ( x 1)3 ( x 1) . Xt hm s f (t ) t 3 t , t , ta c f (t ) 3t 2 1 0, t nn y l hm ng bin. Phng trnh trn c vit li l f ( y ) f ( x 1) y x 1 . Thay vo phng trnh th hai ca h, ta c 1 x 2 x 1 1 x 1 1 x 2 x 1 1 x 1 (*) 47 48. t2 2 . t t x 1 1 x 0 t 2 2 1 x 1 x 2 2Do (*) 22t2 2 t 1 t 2 2t 0 t 2 . 2Khi x 1 1 x 2 2 2 1 x 2 2 x 2 1 x 1 , tha iu kin. Tng ng vi mi gi tr x, ta c y 0, y 2 . Vy h phng trnh cho c hai nghim l ( x, y ) (1, 0), (1, 2) . Nhn xt. bi 1, cch t n ph v phn tch nh th ch mang tnh cht tham kho v n kh thiu t nhin. Ta hon ton c th kho st hm s f ( x ) x 1 x 1 2 x x 2 2 trn[1, 2] , tnh o hm cp 2 chng minh phng trnh f ( x ) 0 c khng qu hai nghim phn bit ri nhm nghim hoc ta cng c th dng phng php nhn lng lin hp gii quyt cng kh thun tin.Bi 40. x 3 8 y 3 4 xy 2 1 1/ Gii h phng trnh 4 4 2 x 8 y 2 x y 0 2/ Chng minh phng trnh sau c ng mt nghim ( x 1) 2011 2( x 1) 3 x 3 3 x 2 3 x 2 . ( d b thi HSG tnh Ngh An) Li gii. 2 2 2 3 3 3 x t xt 1 t 1 x (t x ) 1/ t 2y t , h cho tr thnh 4 4 3 3 4 x t 4 x t 0 4 x ( x 1) t (t 1) 0 Thay t 3 1 x (t 2 x 2 ) t phng trnh th nht vo phng trnh th hai ca h, ta c x 0 4 x( x 3 1) xt (t 2 x 2 ) 0 3 3 2 4( x 1) t tx 0D thy x 0 khng l nghim ca h nn ta ch xt 4( x 3 1) t 3 tx 2 0 4 x 3 t 3 tx 2 4 . Phng trnh th nht ca h c vit li l 4 x3 4t 3 4 xt 2 4 . Do 48 49. 4 x3 4t 3 4 xt 2 4 x 3 t 3 tx 2 t 3 tx 2 t 0 t x . D thy t 0 khng tha mn h cho nn ch xt t x . x 3 x3 x.x 2 1 x3 1 -Nu t x , ta c h 4 4 x 1. 4 4 x x 4 x x 0 5 x 5 x 0 Khi t x 1 y 1 1 , h cho c nghim ( x, y ) (1, ) . 2 22 3 3 3 x x x( x ) 1 x 1 -Nu t x , ta c h 4 4 , h ny v nghim. 4 4 x ( x ) 4 x ( x) 0 5 x 3 x 0 1 Vy h phng trnh cho c nghim duy nht l ( x, y ) (1, ) . 22/ Chng minh phng trnh sau c ng mt nghim ( x 1) 2011 2( x 1) 3 x 3 3 x 2 3 x 2 . iu kin x 1 . Phng trnh cho tng ng vi ( x 1) 2011 2( x 1) 3 ( x 1)3 1 . t t x 1 0 . Ta cn chng minh phng trnh t 2011 2t 3 t 6 1 c ng mt nghim dng. Xt hm s f (t ) t 2011 t 6 2t 3 1, t 0 . Ta c f (0) 1, lim f (t ) v f (t ) lin tc trn x (0, ) nn phng trnh f (t ) 0 c t nht mt nghim dng. Ta c t 2011 2t 3 t 6 1 t 2011 (t 3 1)2 0 t 0 , m vi t 0 , ta c t 2011 (t 3 1)2 1 t 1 . Khi f (t ) 2011t 2010 6t 5 6t 2 1999t 2010 6(t 2010 t 5 ) 6(t 2010 t 2 ) 0 nn y l hm ng bin, tc l n c khng qu mt nghim. Kt hp cc iu ny li, ta thy rng phng trnh t 2011 2t 3 t 6 1 c ng mt nghim dng, tc l phng trnh cho c ng mt nghim. Ta c pcm. Nhn xt. Bi 2 tuy hnh thc kh phc tp nhng qua php t n ph a v mt phng trnh a thc thng thng. tng gii nh trn tng xut hin trong B tuyn sinh ca B GD T, l bi ton sau Chng minh phng trnh x 5 x 2 2 x 1 c ng mt nghim. 49 50. x 3 y x 3 12 Bi 41. Gii h phng trnh sau y 4 z y 3 6 3 9 z 2 x z 32( thi chn i tuyn H KHTN H Ni, vng 1) Li gii. 3 y 6 x 3 x 6 3( y 2) ( x 2)( x 2 2 x 3) H cho tng ng vi 4 z 16 y 3 y 10 4( z 4) ( y 2)( y 2 2 y 5) 3 2 2 x 4 z 9 z 28 2( x 2) ( z 4)( z 4 z 7)Nhn tng v cc phng trnh ca h, ta c 24( x 2)( y 2)( z 4) ( x 2)( y 2)( z 4) ( x 2 2 x 3)( y 2 2 y 5)( z 2 4 z 7) ( x 2)( y 2)( z 4) 0 2 2 2 ( x 2 x 3)( y 2 y 5)( z 4 z 7) 24-Nu ( x 2)( y 2)( z 4) 0 x 2 y 2 z 4 . Ta thy rng nu x 2 th theo phng trnh th nht, y 2 ; theo phng trnh th hai, z 4 v h cho c nghim ( x, y , z ) (2, 2, 4) . Tng t nu y 2 hoc z 4 . -Nu ( x 2 2 x 3)( y 2 2 y 5)( z 2 4 z 7) 24 ( x 1)2 2 ( y 1)2 4 ( z 2)2 3 24 Ta thy rng ( x 1)2 2 ( y 1)2 4 ( z 2)2 3 2.3.4 24 nn ng thc phi xy ra, tc l x 1, y 1, z 2 . Th li, ta thy b ny khng tha mn h cho. Vy h phng trnh cho c nghim duy nht l ( x, y , z ) (2, 2, 4) . Nhn xt. Bi ny cng c dng tng t nh mt bi cp trn v ni chung th dng ny kh quen thuc; tuy nhin, im mi ca bi ny l khng phi nhn c nghim ( x, y , z ) (2, 2, 4) thng qua cc bt ng thc so snh vi nghim na m l qua vic chng minh b nghim qua vic chng minh n l duy nht vi vic dng cc tam thc bc hai. 50 51. y 2 x2 x 2 1 2 e Bi 42. Gii h phng trnh y 1 3 log ( x 2 y 6) 2 log ( x y 2) 1 2 2 ( thi chn i tuyn trng THPT Cao Lnh, ng Thp) Li gii. x 2 y 6 0 iu kin xc nh x y 2 0Xt hm s: f (t ) et (t 1), t [0, ) . Ta c f (t ) et (t 1) et et (t 2) 0 nn y l hm ng bin. Do ey2 x22 2 x2 1 e x ( x 2 1) e y ( y 2 1) f ( x 2 ) f ( y 2 ) x 2 y 2 x y . 2 y 1Phng trnh th hai ca h tng ng vi 3 log 2 ( x 2 y 6) 2 log 2 ( x y 2) 1 log 2 ( x 2 y 6)3 log2 2( x y 2)2 ( x 2 y 6)3 2( x y 2)2 (*) Xt hai trng hp -Nu x y th thay vo (*), ta c (3 x 6)3 2(2 x 2)2 . Theo iu kin ban u th 2 x 2 0 2 x 4 2 x 2 0 . Hn na: (3 x 6)3 2(2 x 4)2 ( x 2)2 (27 x 46) 0 (3x 6)3 2(2 x 4)2 . Do : (3 x 6)3 2(2 x 4)2 2(2 x 2)2 nn phng trnh ny v nghim. -Nu x y , thay vo (*), ta c ( x 6)3 2(2)2 (6 x)3 8 6 x 2 x 4 . Suy ra: y x 4 . Th li thy tha. Vy h cho c nghim duy nht l ( x, y ) (4, 4) . Nhn xt. H dng ny rt quen thuc vi tng chnh l dng tnh cht ca hm n iu: f (a) f (b) a b . bi trn cng ch cc nh gi trong trng hp x = y, bi v khi phng trnh bc ba thu c phi gii theo cng thc tng qut, iu thng b trnh cc k 51 52. thi HSG; do , vic tm mt nh gi thch hp chng minh nghim ca n khng tha bi l mt iu kh t nhin.x2 x 2Bi 43. Gii phng trnh sau:2x2 x 2 x2 11 x x 2 1 x x 4 ( thi HSG tnh Bnh Phc) Li gii. iu kin xc nh 0 x 2 x 2 4, 0 x 2 x 4 1 x 1 17 . 2Phng trnh cho tng ng vix2 x 2 21 4 ( x x 2) Xt hm s f (t ) x2 x 2 x2 11 4 ( x x)t , t 0, 4 , ta c 1 4 t1 4 t t 1 0, t 0, 4 nn y l hm ng bin. f (t ) ( ) 2 t 2 4 t (1 4 t ) 2Phng trnh trn chnh l f ( x 2 x 2) f ( x 2 x) ( x 2 1) 0 (*). Ta xt hai trng hp1 17 x 2 x 2 x 2 x f ( x 2 x 2) f ( x 2 x ) , ng thi x 2 1 0 , 2 khi f ( x 2 x 2) f ( x 2 x) ( x 2 1) 0 . -Nu 1 x -Nu 1 x 1 x 2 x 2 x 2 x f ( x 2 x 2) f ( x 2 x) , ng thi x 2 1 0 , khi f ( x 2 x 2) f ( x 2 x) ( x 2 1) 0 . Th trc tip thy x 1 tha mn (*) Vy phng trnh cho c nghim duy nht l x 1 .52 53. t , t 0, 4 nh trn khng kh, c th thy 1 4 t ngay t vic quan st biu thc v t iu kin xc nh; tuy nhin, vic ny cng d khin ta lm tng n vic xt hm s no m khng ngh ra cch nh gi kiu nh trn.Nhn xt. Vic pht hin ra hm s f (t ) 3Mt bi ton c cng cch nh gi nh trn l e x ( x 3 x) ln( x 2 1) e x . Cc bn th gii thm bi ton saux2 x 2 1 x2 x 2x2 x 1 x2 x 4 x 1.Bi 44. 1/ Gii phng trnh33x 4 x3 3x 2 x 22/ Tm s nghim ca phng trnh (4022 x 2011 4018 x 2009 2 x) 4 2(4022 x 2011 4018 x 2009 2 x) 2 cos 2 2 x 0 ( thi chn i tuyn Chuyn Nguyn Du) Li gii. 1/ Phng trnh cho tng ng vi33x 4 2 x 3 ( x 1) 33 ( x 1) 2 x y 4 t y 1 3x 4 . Ta c h phng trnh 3 ( y 1) 3x 4 3Tr hai phng trnh ca h, v theo v, ta c ( x y ) ( x 1) 2 ( x 1)( y 1) ( y 1) 2 y x x y 0 xy 2 2 ( x 1) ( x 1)( y 1) ( y 1) 1Suy ra x 1 3 3x 4 ( x 1)3 3 x 4 x 3 3x 2 4 ( x 1)( x 2) 2 0 x 1 x 2 . Th li ta thy tha. Vy phng trnh cho c hai nghim phn bit l x 1, x 2 . 2/ t t 4022 x 2011 4018 x 2009 2 x . Ta c53 54. t 2 1 sin 2 x t 2 (sin x cos x ) 2 2 t 4 2t 2 cos 2 2 x 0 (t 2 1)2 sin 2 2 x 2 2 t 1 sin 2 x t (sin x cos x) Ta c bn phng trnh sau t sin x cos x, t (sin x cos x ), t sin x cos x, t sin x cos x . Ta thy hm s t ( x) 4022 x 2011 4018 x 2009 2 x l l nn ch cn xt cc phng trnht ( x) sin x cos x, t ( x ) cos x sin x . Ta c t ( x) sin x cos x 4022 x 2011 4018 x 2009 2 x sin x cos x . Xt hm s g ( x ) 4022 x 2011 4018 x 2009 2 x (sin x cos x) c g ( x ) 4022.2011x 2010 4018.2009. x 2008 2 (cos x sin x) 0 nn l hm ng bin. Hn na g (0) 1, g (1) 0 g (0).g (1) 0 , ng thi g ( x) lin tc trn (0,1) nn phng trnh g ( x) 0 c ng mt nghim thuc (0,1) , tc l phng trnh t ( x) sin x cos x c ng mt nghim thc. Tng t, phng trnh t ( x) cos x sin x cng c ng mt nghim thc thuc (0,1) . Do , mi phng trnh t ( x) cos x sin x v t ( x) cos x sin x cng c mt nghim thc. Vy phng trnh cho c ng 4 nghim thc.(2 x)(1 2 x )(2 y )(1 2 y ) 4 10 z 1 Bi 45. Gii h phng trnh sau 2 2 2 2 2 x y z 2 xz 2 yz x y 1 0 ( thi chn i tuyn H Tnh) Li gii. Ta c x 2 y 2 z 2 2 xz 2 yz x 2 y 2 1 0 ( x y z )2 ( xy 1)2 0 hay 1 1 x y z 0, xy 1 y , z ( x y ) ( x ) . x xThay vo phng trnh th nht ca h, ta c 54 55. 1 2 1 (2 x)(1 2 x )(2 )(1 ) 4 1 10( x ) x x x 1 2x 2 x 1 (2 x)(1 2 x )( )( ) 4 1 10( x ) x x x (4 x 2 )(1 4 x 2 ) 1 1 1 4 1 10( x ) 4( x 2 2 ) 17 4 1 10( x ) 2 x x x xt t x 1 1 t 2 . Ta c t 2 2 x 2 2 , thay vo phng trnh trn, ta c x x4(t 2 2) 17 4 1 10t 4t 2 25 4 1 10t (4t 2 25) 2 16(1 10t ) 0 7 (4t 2 20t 29)(2t 3)(2t 7) 0 t 2 Vi gi tr t ny, ta c x 1 7 7 33 2 x2 7 x 2 0 x . 2 4 x-Vi x 7 7 33 7 33 ,z . , ta tnh c y 4 2 4-Vi x 7 33 7 33 7 , ta tnh c y ,z . 4 4 2Th li ta thy tha. Vy h cho c hai nghim phn bit l( x, y , z ) (7 33 7 33 7 7 33 7 33 7 , , ), ( , , ). 4 4 2 4 4 2Nhn xt. Vic pht hin ra hng ng thc trn l khng kh nhng vic thay cc gi tr vo v tm ra cch t n ph thch hp qu l khng n gin, cn c cch bin i chnh xc. Bi ton c hnh thc v tng cng kh th v. Bi 46. 1/ Gii phng trnh sau 2010 x ( x 2 1 x) 1 . y 4 4 x 2 xy 2 x 4 5 2/ Gii h phng trnh x 3 3 y 2 x y 2 55 56. ( thi chn i tuyn trng THPT So Nam, tnh Qung Nam) Li gii. 1/ Phng trnh cho tng ng vi 2010 x x 2 1 x . Ta s chng minh phng trnh ny c nghim duy nht l x 0 . Tht vy Xt hm s f ( x) 2010 x ( x 2 1 x) , ta c f ( x) 2010 x.ln 2010 (x 2 1)x 1 1-Nu x 0 th f ( x) ln 2010 ( 1) 0 nn y l hm ng bin, m f (0) 0 nn 1 1 2 x phng trnh ny c ng mt nghim x 0 vi x 0 .-Nu x 1 , ta c f ( x ) 2010 x.(ln 2010) 2 13 1 , f ( x) 2010 x.(ln 2010) 3 . 0. 2 2 ( x 1)5 ( x 1) 23Suy ra f ( x ) l hm ng bin nn f ( x ) f (1) (ln 2010)2 1 0 nn f ( x ) l hm 2010 2 2nghch bin, suy ra f ( x) lim f ( x ) lim [2010 x ( x 2 1 x)] 0 nn phng trnh f (0) 0 x x khng c nghim vi x 1 .1 5 1 1 5 1 1 1 th x 2 1 x ( ) 2 1 , 2010 x nn trong 2 2 2 2 2 2010 trng hp ny phng trnh v nghim.-Nu 1 x x 1 1) 0 nn y l hm ng bin, suy ra x 0 th f ( x) 2010 x.ln 2010 ( 2 2 x 1 f ( x) f (0) 0 .-NuTm li, phng trnh cho c nghim duy nht l x 0 . y 4 4 x 2 xy 2 x 4 5 2/ Gii h phng trnh x 3 3 y 2 x y 2 T phng trnh th hai ca h v tnh ng bin ca hm s f (t ) 2t t 3 , ta c x y . 56 57. Thay vo phng trnh th nht ca h, ta cx4 4x 2x22x4 5 5 x 4 4 x 2( x 1)23 8 x4 4x 3 0 ( x 1)2 ( x 2 2 x 3) 0 x 1 Th li, ta thy tha; tng ng vi gi tr x ny, ta c y 1 . Vy h phng trnh cho c nghim duy nht l ( x, y ) (1,1) . x11 xy10 y 22 y12 Bi 47. Gii h phng trnh 4 4 2 2 7 y 13x 8 2 y 3 x(3x 3 y 1) ( thi chn i tuyn TP.HCM) Li gii. Ta thy h ny khng c nghim tha y 0 nn ta ch xt y 0 , khi ta cx x x11 xy10 y 22 y12 ( )11 y11 y . y y Xt hm s f (t ) t11 t , t f (t ) 11t10 1 0, t nn y l hm ng bin.x x ng thc trn chnh l f ( ) f ( y ) y x y 2 . y y Thay vo phng trnh th hai ca h, ta c7 x 2 13 x 8 2 x 2 3 x(3 x 2 3 x 1) t t 7 13 8 3 1 2 3 23 3 2 . x x x x x1 0 . Ta c x7t 13t 2 8t 3 2 3 3 3t t 2 (2t 1)3 2(2 x 1) (3 3t t 2 ) 2 3 3 3t t 2 Xt hm s f (a) a3 2a, a 0 f (a) 3a 2 2 0 nn hm ny ng bin. Phng trnh trn chnh lf (2t 1) f ( 3 3 3t t 2 ) 2t 1 3 3 3t t 2 (2t 1)3 3 3t t 2 (t 1)(8t 2 5t 2) 0 Do t 0 nn khng c gi tr no tha mn. Vy h phng trnh cho v nghim. 57 58. Nhn xt. Do phng trnh thu c sau khi tm ra quan h gia x v y khng c nghim dng nn ta c th dng bt ng thc nh gi thay v dng hm s nh trn. 2009 x 2010 y ( x y ) 2 Bi 48. Gii h phng trnh: 2010 y 2011z ( y z ) 2 2 2011z 2009 x ( z x )( thi chn i tuyn chuyn Quang Trung, Bnh Phc) Li gii. t 2009 a 0 , ta xt h tng qut hn l ax (a 1) y ( x y )2 2 (a 1) y (a 2) z ( y z ) (*) 2 (a 2) z ax ( z x)( x y ) 2 ( z x) 2 ( y z ) 2 Ta tnh c ax ( x y )( x z ) 2 Tng t (a 1) y ( y z )( y z ), (a 2) z ( z x )( z y ) 2T y suy ra ax.(a 1) y.(a 2) z ( x y )( y z )( z x ) 0 Mt khc, t (*) ta thy rng tng ca tng cp trong ba gi tr ax, (a 1) y , (a 2) z u khng m, ta s chng minh rng c ba gi tr ny u khng m. Tht vy, gi s ax 0 x 0 , t phng trnh th nht v phng trnh th ba ca (*), suy ra (a 1) y 0, (a 2) z 0 y , z 0 hay x y, x z 0 ax ( x y )( x z ) 0 , mu thun. Do ax 0 . Tng t, ta cng c (a 1) y , (a 2) z 0 . Nhng tch ca ba s ny li khng m nn ta phi c ax (a 1) y (a 2) z x y z 0 . Th li thy tha. Vy h cho c ng mt nghim l x y z 0 .Nhn xt. R rng cc h s ban u l chn da theo thi quen chn h s trng vi nm cho nn ta hon ton c th xt bi ton tng qut vic bin i thun tin hn. Bi ton thc s th v sau khi c ax ( x y )( x z ), (a 1) y ( y z )( y z ), (a 2) z ( z x)( z y ) . Nu khng dng bt ng thc nh gi m c gng dng cc php th th kh c th thnh cng. 58 59. 1 2 2 x y 5 Bi 49. Gii h phng trnh sau 4 x 2 3 x 57 y (3 x 1) 25 ( thi chn i tuyn Ngh An) Li gii. 10 2 2 5( x 2 y 2 ) 1 2( x y ) 25 H cho tng ng vi 2 57 4 x 3 x 3 xy y 25 2 x 2 2 y 2 3 x 3 xy y 47 25 Ta thy 2 x 2 2 y 2 3 x 3xy y 47 47 (2 x y )( x 2 y ) (2 x y ) ( x 2 y ) 25 25t 2 x y a, 2 x y b , ta c 7 a b 5 12 a 2 b 2 1 (a b) 2 2ab 1 2ab (a b) 2 1 ab 25 47 94 144 2 a b 17 ab a b 2ab 2(a b) (a b 1) 25 25 25 25 ab 132 25 Ta thy h phng trnh th hai v nghim, h th nht c hai nghim l 3 4 4 3 2 1 11 2 (a, b) ( , ), (a, b) ( , ) , tng ng l ( x, y ) ( , ), ( , ) . 5 5 5 5 5 5 25 25 2 1 11 2 Vy h phng trnh cho c hai nghim phn bit l ( x, y ) ( , ), ( , ) . 5 5 25 25Nhn xt. Cch phn tch phng trnh th hai qu tht rt kh thy. Bi ton ny thc cht xut pht t mt h i xng thng thng, nhng qua cc php th v tch biu thc, n tr nn phc tp v vic bin i ngc li thng phi m mn. Ta cng c th nhn phng trnh th nht vi 25 v phng trnh th hai vi 200 ri cng li, ta c 25(3 x y 1)2 144 , cc gi tr 25 v 50 ny chn bng phng php h s bt nh vi mong mun tm ra mt quan h p gia x v y, nh bnh phng trn chng hn. 59 60. Bi 50. Cho cc tham s dng a, b, c . Tm nghim dng ca h phng trnh sau : x y z a b c 2 2 2 4 xyz a x b y c z abc( kim tra i tuyn Ninh Bnh) Li gii.a 2 b 2 c 2 abc 4. Phng trnh th hai ca h tng ng vi yz zx xy xyz t x1 a b c , y1 , z1 , suy ra x12 y12 z12 x1 y1 z1 4 (*). yz zx xyD thy 0 x1 , y1 , z1 2 nn tn ti cc gi tr u, v tha 0 u, v v x1 2 sin u, y1 2sin v . 2Thay vo (*), ta c z12 4 z1.sin u.sin v 4 sin 2 u 4sin 2 v 4 0 . y l phng trnh bc hai theo bin z1 , ta c (2sin u.sin v ) 2 (4 sin 2 u 4 sin 2 v 4) 4(1 sin 2 u )(1 sin 2 v) 4 cos 2 u.cos 2 v 0 . z 2 sin u sin v 2 cos u cos v 0 Suy ra phng trnh ny c hai nghim l 1 z1 2 sin u sin v 2 cos u cos v 0Do a 2 yz .sin u , b 2 zx .sin v, c 2 xy (cos u cos v sin u sin v) . Thay vo phng trnh th nht ca h, ta cx y z 2 yz .sin u 2 zx .sin v 2 xy (cos u cos v sin u sin v) ( x cos v y cos u )2 ( x sin v y sin u z ) 2 0 x cos v y cos u x sin v y sin u z 0 Ta tnh cz x sin v y sin u Tng t, ta cng c y a y a b b x ab z 2 2 zx 2 yz 2 zca bc ,x . 2 260 61. Vy h phng trnh cho c nghim duy nht l ( x, y , z ) (bc ca ab , , ). 2 2 2Nhn xt. y l bi ton trong IMO Shortlist, bi v li gii thc s rt hay, l mt kt hp p gia i s v lng gic. Ta cng c th gii bng bin i i s nh cch t n ph bc ca a b u, y v, z w v nh gi bng bt ng thc. x 2 2 23x y x x2 y 2 3 Bi 51. Gii h phng trnh sau trn tp hp s thc y x 3y 0 x2 y 2 ( thi chn i tuyn Chuyn Vnh Phc, tnh Vnh Phc) Li gii. Ta s gii h phng trnh ny bng s phc. Nhn phng trnh th hai ca h vi i (n v o i 2 1 ) ri cng vi phng trnh th nht, 3 x y xi 3 yi 3( x yi) i( x yi ) 3 ( x yi) 2 2 0 ta c x yi 2 2 x y x y2 x y21 x yi 2 . ng thc trn c vit li l z x y2 3i 3 (1 2i ) 0 z 2 3z 3 i 0 z z 2 i z 1 i . z z 2t z x yi -Nu z 2 i , suy ra x yi 2 i x 2, y 1 . -Nu z 1 i , suy ra x yi 1 i x 1, y 1 . Th li ta thy tha. Vy h cho c hai nghim l ( x, y ) (2,1), (1, 1) . Nhn xt. Dng ton ny cng kh ph bin v u chung tng l gii quyt bng s phc. x 2y 3x 10 y x x2 y 2 1 x x2 y 2 2 , ( chn i tuyn H Ni 2007) Cc bi ton tng t 10 x 3 y 2x y y 2 y 2 0 x2 y 2 x y2 61 62. Trn thc t, ta cng c th gii bng cch dng bin i i s, nhn x v y thch hp vo tng v ca cc phng trnh ri tr li thu c quan h n gin hn gia cc bin ny.Bi 52. Gii h phng trnh: 4 4 x 2x y y 2 2 3 ( x y ) 3 ( kim tra i d tuyn trng THPT Chuyn HSP H Ni) Li gii. t x y a, x y b, 3 c 3 . T phng trnh th hai ca h, ta cTa c x ab 3 c 3 ab c .a b ab ,y . Suy ra 2 2 a b 2 a b 2 ab (a 2 b 2 ) , hn na: x y ( x y )( x y )( x y ) ab 2 2 2 4422(a b) a 3b a c 3b 2 2 2 Do , phng trnh th nht ca h cho tng ng vi 2 x y (a b ) ab 2 a c 3b (a b 2 ) c(a 2 b 2 ) a c 3b 2 2 Ta c h mi l c(a 2 b 2 ) a c 3b c2 c4 c(a 2 2 ) a ca 4 c3 a3 ac 4 (ca 1)(a 3 c 3 ) 0 a a ab c 1 a a c. c 1 Suy ra h ny c hai nghim l (a, b) (c,1);( , c 2 ) . c 62 63. Xt hai trng hp 3 c 1 3 3 1 3 1 ,y . - Nu a c, b 1 th x 2 2 2 1 c3 1 c3 1 1 11 2 11 - Nu a , b c 2 th x c 2 , y c2 3 3 c 2c 2c 2c 2c 3 3 3 3 1 3 3 1 2 1 , , Vy h cho c hai nghim l: ( x, y ) , . 2 2 33 33 Bnh lun. y l mt h phng trnh rt p, hnh thc ca n d lm chng ta bi ri khi khng th nhm c nghim no cng nh tm c mt hm s no kho st nh tng thng thng. Li gii thun ty i s v cch t n ph nh bi cn phi ch , n tng xut hin trong VMO 2005 x 3 3xy 2 49 2 2 x 8 xy y 6 y 17 x 3 1 4 4 x y 4 y 2x Mt bi ton tng t nh trn cng c li gii rt th v ( x 2 y 2 )5 5 0 Bi 53. Gii phng trnh 2 x 2 .sin x x.cos x 3 2 x 1 x 3 x 5 x 1 ( thi chn i tuyn H Ni) Li gii. Ta thy phng trnh khng c nghim x 1 1 nn ta ch xt x . 2 21 Xt hm s f ( x ) 2 x 2 .sin x x.cos x 3 2 x 1 x 5 x 3 x 1, x . 22Ta c f ( x) 3 x.sin x (2 x 2 1) cos x 33 (2 x 1)2 5 x 4 3x 2 1Ta s chng minh nh gi mnh hn l 3 x.sin x (2 x 2 1) cos x 5 x 4 3 x 2 1 0, x (*) 63 64. Ta thy biu thc ny khng thay i khi thay x bi x nn ta ch cn xt x 0 . Ta cn chng minh bt ng thc sau sin x x x3 x2 , cos x 1 , x 0 . 6 2x2 Xt hm s g ( x ) cos x 1 , x 0 , ta c 2 g ( x ) sin x x, g ( x ) cos x 1 0 g ( x ) sin x x g (0) 0 . Do , g ( x) l hm ng bin trn [0, ) , suy ra g ( x ) g (0) 0 cos x 1 Tng t, ta cng c sin x x x2 x2 0 cos x 1 . 2 2x3 . 6T hai nh gi ny, ta c3 x.sin x (2 x 2 1) cos x 5 x 4 3 x 2 1 3 x( x x3 x2 ) (2 x 2 1)(1 ) 5 x 4 3x 2 1 . 6 2Hn na, ta cng cx3 x2 x4 3 2 7 x 4 3x2 2 4 2 2 4 4 2 0 3 x( x ) (2 x 1)(1 ) 5 x 3 x 1 3 x x x 1 5 x 3x 1 6 2 2 2 2 nn 3 x.sin x (2 x 2 1) cos x 5 x 4 3 x 2 1 0, x . Do (*) ng hay f ( x) 0, x . Suy ra f ( x ) l hm ng bin nn phng trnh cho c khng qu mt nghim. Mt khc f (0) 0 nn 0 l nghim ca phng trnh cho. Vy phng trnh cho c nghim duy nht l x 0 . Nhn xt. im quan trng nht ca bi ton l chng minh f ( x ) 0 , nhng l mt biu thc va c cha c sin x, cos x v cn thc, ng thi s hng t do ca hm s li m nn tht s rt kh d on c phi lm g trong trng hp ny. Vic b i biu thc cha cn trn rt quan trng v n gip ta c c mt hm s chn v ch cn xt biu thc trn min [0, ) ;x3 x2 , cos x 1 nn bi ton a v chng 6 2 minh bt ng thc thng thng. Nu khng a cc yu t lng gic v a thc th phi tip tc o hm v cha chc iu ny kh thi. Bt ng thc (*) c th lm mnh thm na l trn min , ta cn c thm hai nh gi sin x x 3 x.sin x (2 x 2 1) cos x 3 4 9 2 x x 1 0, x . 2 2 64 65. ( x 2)2 ( y 3) 2 ( y 3)( x z 2) Bi 54. Gii h phng trnh x 2 5 x 9 z 7 y 15 3 yz 2 2 8 x 18 y 18 xy 18 yz 84 x 72 y 24 z 176( thi chn i tuyn HSP H Ni, ngy 2) Li gii. t a x 2, b y 3 . Thay vo tng phng trnh ca h cho, ta c ( x 2)2 ( y 3)2 ( y 3)( x z 2) a 2 b 2 b(a z 4) a 2 ab b 2 bz 4b 0 , x 2 5 x 9 z 7 y 15 3 yz a 2 a 7b 3bz 08 x 2 18 y 2 18 xy 18 yz 84 x 72 y 24 z 176 8a 2 2a 18b 2 72b 18ab 18bz 30 z 94 0 8a 2 2a 18(b2 ab bz 4b) 30 z 94 0 a 2 ab b 2 bz 4b 0 Suy ra a 2 a 7b 3bz 0 (*) 2 2 8a 2a 18(b ab bz 4b) 30 z 94 0T phng trnh th nht v phng trnh th ba, ta c5a 2 a 47 8a 2a 18a 30 z 94 0 10a 2a 30 z 94 0 z . 15 222Thay vo phng trnh th hai, ta c 5a 2 a 47 5a 2 a 12 5(a 2 a) . a 2 a 7b b 0 b a2 a b 2 5 5 5a a 12 Nhn phng trnh th nht ca h (*) vi 3 ri tr cho phng trnh th hai, ta c2a 2 a 3ab 3b 2 5b 0 Thay z 5(a 2 a ) 5a 2 a 47 v b 2 vo phng trnh ny, ta c 5a a 12 1565 66. 2 5(a 2 a ) 15a (a 2 a ) 25(a 2 a) 3 2 2 0 2a a 2 5a a 12 5a a 12 5a a 12 2 (2a 2 a )(5a 2 a 12)2 15a (a 2 a ) 25(a 2 a ) (5a 2 a 12) 75( a 2 a ) 2 0 50a 6 70a5 208a 4 94a3 482a 2 156a 0 a (a 2)(5a 2 14a 13)(5a 2 11a 3) 0 a 0 a 2 a 11 61 10Tng ng vi cc gi tr ny, ta tm c bn nghim ca h cho l ( x, y , z ) (2, 3, (47 31 61 2 61 28 13 61 4 29 ), ( 4, , ), ( , , ), 15 3 15 10 15 1561 31 2 61 28 39 61 , , ) 10 15 15Nhn xt. Vic t n ph a x 2, b y 3 lm cho h cho n gin i kh nhiu nhng cc lin h phc tp gia cc bin th vn cn. Bi ton y c th c gii theo mt cch nhn cc phng trnh cho mt i lng thch hp ri cng li nhng r rng iu ny khng phi d dng thc hin c. Vic dng php th tuy phc tp nhng li rt t nhin v cng may mn l phng trnh cui khng c cha cn g na. y tnh ton kh nng v cng khng d dng m t tin bin i biu thc nhn c sau php th khi cha chc g n c nghim p m nh gi.Bi 55. 2 z ( x y ) 1 x 2 y 2 Tm x, y, z tha mn h y 2 z 2 1 2 xy 2 zx 2 yz 2 2 y (3x 1) 2 x ( x 1)( thi chn i tuyn trng H KHTN H Ni, vng 3) Li gii. T phng trnh th ba ca h, ta c(3x 3 x) 2 x ( x 2 1) x3 3x 2 x( x 2 1) y x y x y 2 . (3x 2 1) (3x 2 1) 3x 1 t x tan , (tan 3 3 tan , ) cos 0 . Ta c tan y y tan 3 tan . 3tan 2 1 2 2 66 67. T phng trnh th nht ca h, ta c x 2 y 2 1 (2 tan tan 3 ).tan 3 1 2 tan .tan 3 tan 2 3 1 z 2( x y ) 2 tan 3 2 tan 3 tan 3 cot 3 1 sin 3 cos 3 1 tan tan ( ) tan 2 2 cos 3 sin 3 sin 6 T phng trnh th hai ca h, ta c x 2 y 2 z 2 2 xy 2 zx 2 yz 1 x 2 ( y z x)2 1 x 2 (tan 3 tan tan 1 tan ) 2 1 tan 2 sin 6sin 3 1 1 2sin 2 3 1 1 tan ) 2 ( tan ) 2 2 cos 3 2 sin 3 cos 3 cos 2sin 3 cos 3 cos 2 cos 6 1 cos 6 cos sin 6 sin 2 1 ) ( tan )2 ( 2 sin 6 cos sin 6 cos cos 2 cos 5 1 )2 ( cos 5 cos sin 6 cos cos 5 cos( 6 ) 2 sin 6 cos cos 2( k 2 22 11 , 2 k 2 cos 5 cos( 2 6 ) 5 ( 2 6 ) k 2 ,k k 2 , k 2 cos 5 cos( 6 ) 5 ( 6 ) k 2 2 2 22 11 2 Do ( , ) nn hai h nghim k 2 , k khng tha mn. 2 2 2 k 2 , ta tm c tt c 10 gi tr tha mn l 22 11 3 5 7 9 , , , , . 22 22 22 22 22Vi hai h nghim Vy h phng trnh cho c cc nghim l( x, y , z ) (tan , tan 3 tan , tan 1 3 5 7 9 ), , , , , . sin 6 22 22 22 22 22Nhn xt. Cch dng lng gic y c l l con ng duy nht gii bi ny bi v vi cc nghim nh trn th khng th c cch i s no m tm ra c. tng quan trng nht ny xut pht t biu thc ca x y hon ton ging h s ca khai trin tan 3 . Do , bi ny tuy bin i phc tp nhng tng cng kh t nhin!67